Answer:
[tex]\Omega = \{ 1HH, 1HT, 1TH, 1TT, 2H, 2T, 3HH, 3HT, 3TH, 3TT, 4H, 4T,\\5HH , 5HT, 5TH, 5TT, 6H, 6T \}[/tex]
The probability is 3/12. The third option is correct.
Step-by-step explanation:
The sample space is
[tex]\Omega = \{ 1HH, 1HT, 1TH, 1TT, 2H, 2T, 3HH, 3HT, 3TH, 3TT, 4H, 4T,\\5HH , 5HT, 5TH, 5TT, 6H, 6T \}[/tex]
Note that this sample space is not equally probable.
The probability of getting a given number followed is the probability of getting an even number from the 6 numbers (3/6) multiplied by the probability of getting a head after getting that even number, that is 1/2, because is equally probable to get heads or tails from one single coin toss (note that we are assuming that the dice was even, thats why there is a single coin toss).
Therefore, the probability of getting an even number and a head is
P( D in {2,4,6} , H = 1) = P(D in {2,4,6}) * P(H=1 | D in {2,4,6}) = 3/6 * 1/2 = 3/12.
The sample space S can be constructed by listing all possible outcomes. The probability of getting an even number on the die followed by one head is 1/4.
Explanation:The sample space S can be constructed by listing all possible outcomes. Since there are 6 possible outcomes for the die and 2 possible outcomes for the coin flip, the total number of outcomes is 6 * 2 = 12. The sample space S is {1HH, 1HTT, 2H, 2HTT, 3HH, 3HT, 4H, 4HTT, 5HH, 5HTT, 6H, 6HTT}.
The probability of getting an even number on the die followed by one head can be found by counting the number of favorable outcomes (even number on the die followed by one head) and dividing it by the total number of outcomes. From the sample space, we can see that there are 3 favorable outcomes: 2H, 4H, and 6H. Therefore, the probability is 3/12, which can be simplified to 1/4.
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Confirm that f and g are inverses by showing that f(g(x)) = x and g(f(x)) = x. (5 points)
f(x) = quantity x minus nine divided by quantity x plus five. and g(x) = quantity negative five x minus nine divided by quantity x minus one.
the reason that these are inverses is because if you switch x with z
and g(f(x))=x, then (x-9)/(x+5) is (z-9)/(z+5)=x. which implies that:
(z+5) - 14/(z+5)=1-(14/(z-5))=x.
(1/(1 - x) * 14) + 5 = z = g(x) maybe i messed up?
Step-by-step explanation:
f(x) = (x − 9) / (x + 5)
g(x) = (-5x − 9) / (x − 1)
To find f(g(x)), substitute g(x) into f(x).
f(g(x)) = [(-5x − 9) / (x − 1) − 9] / [(-5x − 9) / (x − 1) + 5]
Multiply top and bottom by x − 1.
f(g(x)) = [(-5x − 9) − 9(x − 1)] / [(-5x − 9) + 5(x − 1)]
Simplify.
f(g(x)) = (-5x − 9 − 9x + 9) / (-5x − 9 + 5x − 5)
f(g(x)) = (-14x) / (-14)
f(g(x)) = x
To find g(f(x)), substitute f(x) into g(x).
g(f(x) = [-5(x − 9) / (x + 5) − 9] / [(x − 9) / (x + 5) − 1]
Multiply top and bottom by x + 5.
g(f(x) = [-5(x − 9) − 9(x + 5)] / [(x − 9) − (x + 5)]
Simplify.
g(f(x) = (-5x + 45 − 9x − 45) / (x − 9 − x − 5)
g(f(x) = (-14x) / (-14)
g(f(x) = x
The width of a rectangle, in feet, is represented by left parenthesis 3 x minus 1.5 right parenthesis. The length of the rectangle, in feet, is represented by left parenthesis 1.25 x plus 3 right parenthesis. Find the perimeter of the rectangle.
Given the width and length of the rectangle in terms of x, the formula for the perimeter is substituted with these expressions. After simplifying, the perimeter of the rectangle is found to be represented by the equation 8.5x + 3.
Explanation:The subject represents a problem in mathematics, specifically geometry. The perimeter of a rectangle is calculated by the formula 2(width + length). Given that the width is meant to be represented by (3x - 1.5) and the length is represented by (1.25x + 3), we substitute these expressions into our formula to find the perimeter of the rectangle.
Perimeter = 2[(3x - 1.5) + (1.25x + 3)].
To simplify, the above equation becomes:
Perimeter = 2[4.25x + 1.5],
which then further simplifies to:
Perimeter = 8.5x + 3.
The perimeter of the rectangle is therefore represented by the expression 8.5x + 3.
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Simplify. 2( 5 3 + 3 4 ) − 4 3
Each month, Daddy gives barbara $50 . From that barbara saves $20 each month and spends the rest . How much money would barbara have spent in 5 months
Answer:
$150
Step-by-step explanation:
Dad gives the girl 50 and she save 20 so 50-20 = 30. Then u multiply that by the number of months given...so 30*5= 150
Answer: 150
Step-by-step explanation:
50-20=30
30x5=150
A college infirmary conducted an experiment to determine the degree of relief provided by three cough remedies. Each cough remedy was tried on 50 students and the accompanying data recorded. Test the hypothesis that the three cough remedies are equally effective. Use a P-value in your conclusion.
Answer:
[tex]p_v = P(\chi^2_{4,0.05} >3.81)=0.43233[/tex]
Since the p values is higher than the significance level we FAIL to reject the null hypothesis at 5% of significance, and we can conclude that we don't have significant differences between the 3 remedies analyzed. So we can say that the 3 remedies ar approximately equally effective.
Step-by-step explanation:
A chi-square goodness of fit test "determines if a sample data matches a population".
A chi-square test for independence "compares two variables in a contingency table to see if they are related. In a more general sense, it tests to see whether distributions of categorical variables differ from each another".
Assume the following dataset:
NyQuil Robitussin Triaminic Total
No relief 11 13 9 33
Some relief 32 28 27 87
Total relief 7 9 14 30
Total 50 50 50 150
We need to conduct a chi square test in order to check the following hypothesis:
H0: There is no difference in the three remedies
H1: There is a difference in the three remedies
The level os significance assumed for this case is [tex]\alpha=0.05[/tex]
The statistic to check the hypothesis is given by:
[tex]\sum_{i=1}^n \frac{(O_i -E_i)^2}{E_i}[/tex]
The table given represent the observed values, we just need to calculate the expected values with the following formula [tex]E_i = \frac{total col * total row}{grand total}[/tex]
And the calculations are given by:
[tex]E_{1} =\frac{50*33}{150}=11[/tex]
[tex]E_{2} =\frac{50*33}{150}=11[/tex]
[tex]E_{3} =\frac{50*33}{150}=11[/tex]
[tex]E_{4} =\frac{50*87}{150}=29[/tex]
[tex]E_{5} =\frac{50*87}{150}=29[/tex]
[tex]E_{6} =\frac{50*87}{150}=29[/tex]
[tex]E_{7} =\frac{50*30}{150}=10[/tex]
[tex]E_{8} =\frac{50*30}{150}=10[/tex]
[tex]E_{9} =\frac{50*30}{150}=10[/tex]
And the expected values are given by:
NyQuil Robitussin Triaminic Total
No relief 11 11 11 33
Some relief 29 29 29 87
Total relief 10 10 10 30
Total 50 50 50 150
And now we can calculate the statistic:
[tex]\chi^2 = \frac{(11-11)^2}{11}+\frac{(13-11)^2}{11}+\frac{(9-11)^2}{11}+\frac{(32-29)^2}{29}+\frac{(28-29)^2}{29}+\frac{(27-29)^2}{29}+\frac{(7-10)^2}{10}+\frac{(9-10)^2}{10}+\frac{(14-10)^2}{10} =3.81[/tex]
Now we can calculate the degrees of freedom for the statistic given by:
[tex]df=(rows-1)(cols-1)=(3-1)(3-1)=4[/tex]
And we can calculate the p value given by:
[tex]p_v = P(\chi^2_{4,0.05} >3.81)=0.43233[/tex]
And we can find the p value using the following excel code:
"=1-CHISQ.DIST(3.81,4,TRUE)"
Since the p values is higher than the significance level we FAIL to reject the null hypothesis at 5% of significance, and we can conclude that we don't have significant differences between the 3 remedies analyzed.
To test the hypothesis that three cough remedies are equally effective, conduct an Analysis of Variance (ANOVA) test and use the p-value, compared to the pre-set level of 0.05, to decide if the null hypothesis should be rejected or failed to reject.
Explanation:The subject matter pertains to hypothesis testing, a method used in statistics to test the validity of a claim (hypothesis) about a population. In this case, the null hypothesis in your question is that the three cough remedies are equally effective. The alternative hypothesis (Ha) is that at least one of the remedies is different. The P-value, when set at 0.05, helps to decide on the rejection or non-rejection of the null hypothesis, depending on whether it’s greater or smaller than the P-value.
First, we need to perform an Analysis of Variance (ANOVA) test since we have more than two samples to compare. After conducting the test, we compare the P-value with our pre-set alpha (0.05). If the p-value obtained is less than 0.05, we can reject the null hypothesis, letting us conclude that not all cough remedies are equally effective. If, however, the p-value is more than 0.05, we fail to reject the null hypothesis, and hence, we can't confidently say that one cough remedy is more effective than the others.
Remember that failing to reject the null hypothesis does not prove it true, it only suggests that there's not enough evidence against it given our data and chosen significance level.
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A set is _____ under an operation (such as addition, etc) if any elements of that set always generate element IN set, when the operation is performed on them
Answer:
"closed"
Step-by-step explanation:
A set is closed with respect to an operation if the operation on members of the set produces a member of the set.
Firefighters dig a triangular trench around a forest to prevent the fire from spreading. Two of the trenches are 800 m long and 650 m long. the angle between them is 30°. Determine the area that is enclosed by these trenches.
Answer:
130000m^2
Step-by-step explanation:
a = 800m
b = 800m
c= 650m
α = 30°
4th he triangular trench is an isosceles triangle.
Area of a triangle = 1/2(bcsinA)
= 1/2(800*650*sin30°)
= 130,000m^2
Answer:
The area enclosed by the trenches is 130 000[tex]m^{2}[/tex]
Step-by-step explanation:
The two given sides of the trenches are 800m and 650m. Since the included angle of the two sides are given, then the area covered by the trenches can be calculated using the formula;
Area of a triangle = [tex]\frac{1}{2}[/tex] × abSin C
where: a is the length of one side and b the length of the second side and C represents the value of the included angle.
Thus,
a = 800m, b = 650m and C = 30°
So that,
Area enclosed by the trenches = [tex]\frac{1}{2}[/tex] × abSin C
= [tex]\frac{1}{2}[/tex] × 800 × 650 × Sin30°
= [tex]\frac{1}{2}[/tex] × 800 × 650 × 0.5
= 130 000
The area enclosed by the trenches is 130 000[tex]m^{2}[/tex].
Elise is making necklaces. She has 15 purple beads and 12 yellow beads. If Elise wants to make all the necklaces exactly the same with no beads left over, what is the greatest number of necklaces she can make?
Answer:
3 necklaces.
Step-by-step explanation:
According to the Question,
Elise is having 15 Purple beads and 12 Yellow beads ,
All necklace should be the same ,
So, She wants to form group such that in each group , equal number of Purple and Yellow beads will be present.
Now , the maximum number of such necklaces can be calculated from the HCF of 15 and 12 as we want to break them in equal groups .
Now , HCF of 15 and 12 comes out to be 3.
Thus ,
a maximum of 3 necklaces can be made.
If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the point in the distribution in which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot.
A. 2.8 minutes
B. 3.2 minutes
C. 3.4 minutes
D. 4.2 minutes
Answer:
A. 2.8 minutes
Step-by-step explanation:
1) Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
2) Solution to the problem
Let X the random variable that represent the length of time it takes a college student to find a parking spot in the library parking of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(3.5,1)[/tex]
Where [tex]\mu=3.5[/tex] and [tex]\sigma=1[/tex]
And we need to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.758[/tex] (a)
[tex]P(X<a)=0.242[/tex] (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.242 of the area on the left and 0.758 of the area on the right it's z=-0.700. On this case P(Z<-0.700)=0.242 and P(z>-0.700)=0.758
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.242[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.242[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=-0.700<\frac{a-3.5}{1}[/tex]
And if we solve for a we got
[tex]a=3.5 -0.700*1=2.8[/tex]
So the value of height that separates the bottom 24.2% of data from the top 75.8% is 2.8 minutes.
The point in time where 75.8% of students exceed when finding a parking spot is calculated using Z-scores in a normal distribution. The time is approximately 2.8 minutes.
Explanation:The process to find the point in the distribution where 75.8% of the students exceed the required time to find a parking spot in the library parking lot is essentially to calculate the Z-score corresponding to a given cumulative probability (or area) in the standard normal curve. If the area to the left of the Z-score is 75.8%, then the area to the right of the Z-score is 100%-75.8% = 24.2%.
We then find the Z-score corresponding to 24.2% in the Z-tables. In this case, the corresponding Z-score is approximately -0.7 (since the tables usually give the area to the left, and we are dealing with the area to the right of the Z-score, we consider the negative of the found Z-score).
To convert this back to the time it takes to find a parking spot, we use the formula for converting Z-scores back to raw scores: X = μ + Zσ, where X is the raw score (time to find a parking spot), μ is the mean time to find a parking spot, Z is the Z-score, and σ is the standard deviation of the time to find a parking spot.
Substituting the given values, we get X = 3.5 + (-0.7)*1 = 2.8.
Thus, the time at which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot is 2.8 minutes (option A).
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solve
5x + y = 13
3x = 15 – 3y
Answer:
Solution
(x, y) = (2, 3)
Step-by-step explanation:
Zachary and his children went into a bakery and he bought $15 worth of cupcakes and brownies. Each cupcake costs $3 and each brownie costs $1.50. He bought 3 times as many brownies as cupcakes. Determine the number of cupcakes and the number of brownies that Zachary bought.
Step-by-step explanation:
the answer is
Y=2.5
X=3.75
Answer:
6 brownies and 2 cupcakes.
Step-by-step explanation:
Let the number of brownies be b and the number of cupckes be c.
We have the following system:
b = 3c
1.5b + 3c = 15
Substituting for b in the second equation:
4.5c + 3c = 15
7.5c = 15
c = 2.
Thus b = 3c = 6.
What point located in quadrant I has an x-value that is 2 units from the origin and a y-value 7 units from the origin?
Final answer:
The point in Quadrant I with an x-value 2 units from the origin and a y-value 7 units from the origin has coordinates (2, 7).
Explanation:
The student is asked to determine the coordinates of a point that lies in Quadrant I of the Cartesian coordinate system, with an x-value that is 2 units from the origin and a y-value 7 units from the origin.
The Cartesian coordinate system is divided into four quadrants, and Quadrant I is where both x and y values are positive.
Therefore, a point located 2 units away from the origin on the x-axis, and 7 units away on the y-axis, would have coordinates (2, 7).
Remember, the origin is the point where the x-axis and y-axis intersect, (0,0). Moving 2 units to the right, you reach an x-value of 2. Then, moving up 7 units, you achieve a y-value of 7.
Thus, the coordinates are (2, 7).
If q workers can paint a house in d days, how many days will it take q+2 workers to paint the same house, assuming all workers paint at the same rate ?A. d+2B. d-2C. q+2 / qdD. qd / q+2E. (qd + 2d) / q
Answer: D. [tex]\dfrac{qd}{(q+2)}[/tex]
Step-by-step explanation:
Given : q workers can paint a house in d days.
Let [tex]d_1[/tex] be the number of days taken by q+2 workers to paint the same house.
Since there is inverse relationship between the number of workers and the number of days to do same work ( condition - all workers paints at the same rate), as the number of workers increases the number of days to complete it decreases.
Equation of inverse variation between x and y : [tex]x_1y_1=x_2y_2[/tex]
Substitute , [tex]x_1=q ,\ y_1=d[/tex] and [tex]x_2=q+2 ,\ y_2=d_2[/tex] , we get
[tex]qd=(q+2)d_2\\\\\Rightarrow\ d_2=\dfrac{qd}{(q+2)}[/tex]
Therefore , the number of days it will take q+2 workers to paint the same house = [tex]\dfrac{qd}{(q+2)}[/tex]
Hence, the correct answer is : D. [tex]\dfrac{qd}{(q+2)}[/tex]
If x, y, and z are integers greater than 1, and (327)(510)(z) = (58)(914)(xy), then what is the value of x?
(1) y is prime
(2) x is prime
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
Answer:
Statements (1) and (2) TOGETHER are NOT sufficient.
Explanation:
As in the equation (327)(510)(z) = (58)(914)(xy) there are THREE variables in total i.e. "x", "y" and "z" hence minimum three equations are required to find out values of all variables. Hence,
If the given number of equations is equal to total variable used in any of the equation, values of all the variables can be find out otherwise there can be unlimited number of solutions.
So, value of "x" cannot be determined with the given data.
1. Question: What is the number of the parking space covered by the car?
20 seconds to solve the problem!
will get brainiest ;b
Answer: the number of the parking space covered by the car is 87
Step-by-step explanation:
Numbers are assigned to each parking spot. Looking closely at the numbers assigned to each spot, the numbers are inverted and the number on each successive spot differ by one. The numbers are 86, 87, 88, 89, 90, 91
Therefore, the number assigned to the spot where the car would be 87
You want to paddle a canoe across a small lake and want to know how far it is to the other side. You take measurements ln your side of the lake and make the drawing shown . What is the distance x across the lake
Answer:
[tex] x= 5*400 ft = 2000 ft[/tex]
Step-by-step explanation:
For this case we can use the figure attached and we are interested in order to find the value of x.
We have two similar triangles (DEC and ABC) and we can find the scale factor like this:
[tex]Factor= \frac{EC}{BC}=\frac{500ft}{100ft}=5[/tex]
And now we can apply proportions in order to find the value of x using the two sides DE and BA, since we have the ratio between the triangle DEC and ABC we have this:
[tex]Factor=5=\frac{x ft}{400 ft}[/tex]
And solving for x we got:
[tex] x= 5*400 ft = 2000 ft[/tex]
And then the distance across the lake would be 2000 ft
Answer:
ewq
Step-by-step explanation:
The total number of relative maximum and minimum points of the function whose derivative is f ' (x) = x2(x + 1)3(x – 4)3 is (A) 0 (B) 1 (C) 2 (D) 3 (E) 4 ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 12. Find all absolute and relative
The function's derivative, f'(x) = x²(x + 1)³3(x - 4)³, has 2 relative maximum and minimum points.
So, the correct answer is C) 2.
Explanation:The function's derivative, f'(x) = x²(x + 1)³3(x - 4)³, gives us information about the critical points of the function. Relative maximum and minimum points occur where the derivative is zero or undefined. To find these points, we set the derivative equal to zero and solve for x: x²(x + 1)³3(x - 4)³ = 0. By analyzing the signs of the factors, we can determine the number of relative maximum and minimum points:
When x = 0, both x² and (x + 1)³ are negative, while (x - 4)³ is positive. So, this point is a relative maximum.When x = -1, both x² and (x - 4)^3 are negative, while (x + 1)³ is positive. So, this point is a relative minimum.When x = 4, both (x + 1)³ and (x - 4)³ are positive, while x² is zero. As (x + 1)³and (x - 4)³ are both cubed, this point represents a saddle point rather than a relative maximum or minimum.Therefore, the total number of relative maximum and minimum points is 2.
So, the correct answer is C) 2.
Accounts Payable 420 Accounts Receivable 3,200 Capital Stock 240 Cash 100 Cost of Goods Sold 600 Inventory 380 Long-term Debt 4,640 Net Income 280 Property, Plant, and Equipment (net) 1,400 Retained Earnings (220) Sales 3,000 Note: The retained earnings amount reported is as of the END of the year (after the closing entries have been made). The number of shares outstanding is 100. Compute BOOK VALUE PER SHARE.
Answer:
??????????????????? what
This should be easy for you
Please help me
Explanation:
a. The line joining the midpoints of the parallel bases is perpendicular to both of them. It is the line of symmetry for the trapezoid. This means the angles and sides on one side of that line of symmetry are congruent to the corresponding angles and sides on the other side of the line. The diagonals are the same length.
__
b. We observe that adjacent pairs of points have the same x-coordinate, so are on vertical lines, which have undefined slope. KN is a segment of the line x=1; LM is a segment of the line x=3. If the trapezoid is isosceles, the midpoints of these segments will be on a horizontal line. The midpoint of KN is at y=(3-2)/2 = 1/2. The midpoint of LM is at y=(1+0)/2 = 1/2. These points are on the same horizontal line, so the trapezoid is isosceles.
__
c. We observed in part (b) that the parallel sides are KN and LM. The coordinate difference between K and L is (1, 3) -(3, 1) = (-2, 2). That is, segment KL is the hypotenuse of an isosceles right triangle with side lengths 2, so the lengths of KL and MN are both 2√2.
_____
For part (c), we used the shortcut that the hypotenuse of an isosceles right triangle is √2 times the leg length.
Given points A (1, 2/3), B (x, -4/5), and C (-1/2, 4) determine the value of x such that all three points are collinear
Answer:
[tex]x=\frac{83}{50}[/tex]
Step-by-step explanation:
we know that
If the three points are collinear
then
[tex]m_A_B=m_A_C[/tex]
we have
A (1, 2/3), B (x, -4/5), and C (-1/2, 4)
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
step 1
Find the slope AB
we have
[tex]A(1,\frac{2}{3}),B(x,-\frac{4}{5})[/tex]
substitute in the formula
[tex]m_A_B=\frac{-\frac{4}{5}-\frac{2}{3}}{x-1}[/tex]
[tex]m_A_B=\frac{\frac{-12-10}{15}}{x-1}[/tex]
[tex]m_A_B=-\frac{22}{15(x-1)}[/tex]
step 2
Find the slope AC
we have
[tex]A(1,\frac{2}{3}),C(-\frac{1}{2},4)[/tex]
substitute in the formula
[tex]m_A_C=\frac{4-\frac{2}{3}}{-\frac{1}{2}-1}[/tex]
[tex]m_A_C=\frac{\frac{10}{3}}{-\frac{3}{2}}[/tex]
[tex]m_A_C=-\frac{20}{9}[/tex]
step 3
Equate the slopes
[tex]m_A_B=m_A_C[/tex]
[tex]-\frac{22}{15(x-1)}=-\frac{20}{9}[/tex]
solve for x
[tex]15(x-1)20=22(9)[/tex]
[tex]300x-300=198[/tex]
[tex]300x=198+300[/tex]
[tex]300x=498[/tex]
[tex]x=\frac{498}{300}[/tex]
simplify
[tex]x=\frac{83}{50}[/tex]
Help with this exercise
Answer:
View Image
Step-by-step explanation:
Solve for y.
You have a ≥ so it's a solid line and you shade above that line.
PLZ HELP ASAP 15 pts
Which of the following tables represents a function?
x 4 4 9 9
y 2 −2 3 −3
x 2 −2 3 −3
y 4 4 9 −9
x 1 −1 1 −1
y 4 5 6 7
x 5 5 6 6
y 1 2 3 4
Answer:
x 2 −2 3 −3
y 4 4 9 −9
Step-by-step explanation:
A table does not represent a function if any x-value is repeated. The table shown above is the only one with unique x-values.
A city planner is mapping out some new features for a triangle park. She sketches the park on a grid paper. The coordinates of the vertices of the park are (1, 1), (1, 4), and (-3, 4).
The coordinates provided represent vertices of a triangle on a 2-dimensional grid like a city map. Visualizing the points and connecting them creates a right triangle. Vectors can express the sides of this triangle in terms of magnitude and direction.
Explanation:This is a problem pertaining to geometry and the use of coordinate systems. In the question, we're given vertices of a triangle on a coordinate grid. The coordinates, (1, 1), (1, 4), and (-3, 4), denote specific points on this two-dimensional grid system. Just as in a city map, these coordinates help define the exact location of the points in a well-defined and universally understood manner.
To visualize the triangle formed by these coordinates on a city map, we can consider (1, 1) as a starting point. From this point, move 3 units north to reach (1, 4), then move 4 units to the left to reach (-3, 4), and finally move back 3 units south to reach the starting point (1, 1). This forms a right triangle, with (1, 4) as the right-angle vertex.
Similar to the concept of directional travel mentioned in the supplementary text, vectors can also be utilized in such coordinate systems to describe movement or distance between different points on the map. In our triangle example, each side of the triangle can be represented by a vector, with its magnitude (length) and direction.
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To find the area of the triangle park, we can calculate the lengths of the sides using the distance formula and then use Heron's formula to find the area. The area of the triangle park is 6 square units.
Explanation:The given question is about a city planner mapping out a triangle park on a grid paper using coordinates. The vertices of the park are given as (1, 1), (1, 4), and (-3, 4). The subject of this question is Mathematics, and the grade level is High School.
To find the area of the triangle park, we can use the formula for the area of a triangle. We connect the vertices with line segments and calculate the lengths of the sides. Then, using the Heron's formula, we can calculate the area.
First, we calculate the lengths of the sides using the distance formula. The length of AB is sqrt((1-1)^2 + (4-1)^2) = sqrt(9) = √9 = 3. The length of BC is sqrt((1-(-3))^2 + (4-4)^2) = sqrt(16) = √16 = 4. The length of AC is sqrt((1-(-3))^2 + (4-1)^2) = sqrt(25) = √25 = 5.
Next, we use Heron's formula to calculate the area. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by: Area = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (s = (a+b+c)/2).
In this case, a = 3, b = 4, and c = 5. Therefore, s = (3+4+5)/2 = 12/2 = 6. Plugging in the values, the area of the triangle is given by: Area = sqrt(6(6-3)(6-4)(6-5)) = sqrt(6*3*2*1) = sqrt(36) = √36 = 6.
Therefore, the area of the triangle park is 6 square units.
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he number of E.coli bacteria cells in a pond of stagnant water can be represented by the function below, where A represents the number of E.coli bacteria cells per 100 mL of water and t represents the time, in years, that has elapsed.
Based on the model, by approximately what percent does the number of E.coli bacteria cells increase each year?
A.
59%
B.
40%
C.
41%
D.
60%
The function that represents the number of E.coli bacteria cells per 100 mL of water as t years elapses, and is missing in the question, is:
[tex]A(t)=136(1.123)^{4t}[/tex]
Answer:
Option A. 59%
Explanation:
The base, 1.123, represents the multiplicative constant rate of change of the function, so you just must substitute 1 for t in the power part of the function::
[tex]A(t)=136(1.123)^{4t}[/tex] [tex]rate=(1.123)^{4t}=(1.123)^4=1.590[/tex]Then, the multiplicative rate of change is 1.590, which means that every year the number of E.coli bacteria cells per 100 mL of water increases by a factor of 1.590, and that is 1.59 - 1 = 0.590 or 59% increase.
You can calculate it also using two consecutive values for t. For instance, use t =1 and t = 1
[tex]t=1\\\\ A(1)=136(1.123)^4\\\\\\t=2\\ \\ \\A(2)=136(1.123)^8\\ \\ \\A(2)/A(1)=1.123^8/1.1123^4=1.123^4=1.590[/tex]
The combined ages of a dog and his owner are 96 years in total. The owner is 3 times older than his dog. How old is the owner?
Answer:
72 yrs old.
Step-by-step explanation:
The combined age
D+O=96
The owner is 3 times older than dog
O=3D
D+3D=96
4D = 96
D= 24
Now substitute the value of D in O=3D
O=3.24= 72
The owner is 72 years old.
To solve this problem, we can use algebra. Let's define the dog's age as 'd' and the owner's age as 'o'. By solving the two equations (o + d = 96 and o = 3d), we can determine that the dog is 24 years old and the owner is 72 years old.
Explanation:The question asks us to find the age of a dog's owner, given that the combined ages of the owner and the dog are 96 years, and the owner's age is three times the age of the dog. We can use algebra to solve this problem.
Let's define the dog's age as 'd' and the owner's age as 'o'. We know that o = 3d (the owner's age is three times the dog's age) and o + d = 96 (the combined ages of the owner and the dog are 96).
To find the owner's age, substitute '3d' for 'o' in the second equation: 3d + d = 96. This simplifies to 4d = 96. Dividing both sides of the equation by 4 gives us d = 24, meaning the dog is 24 years old. Now we substitute d = 24 into the equation o = 3d, resulting in o = 72. Therefore, the owner is 72 years old.
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Which product will result in a sum or difference of cubes? (x + 7)(x2 – 7x + 14) (x + 8)(x2 + 8x + 64) (x – 9)(x2 + 9x + 81) (x – 10)(x2 – 10x + 100)
Answer:
Option 3)
[tex]x^3-9^3 = (x-9)(x^2 + 9x + 81)[/tex]
Step-by-step explanation:
We use the identities:
[tex]a^3 + b^3 = (a+b)(a^2-ab+b^2)\\a^3-b^3 = (a-b)(a^2+ab+b^2)[/tex]
1.
[tex](x + 7)(x^2 -7x + 14)[/tex]
It is not a sum or difference of cubes because it does not satisfies the identity.
2.
[tex](x + 8)(x^2 + 8x + 64)[/tex]
It is not a sum or difference of cubes because it does not satisfies the identity.
3.
[tex](x - 9)(x^2 + 9x + 81)\\\text{Comparing with the identity:} \\a^3-b^3 = (a-b)(a^2+ab+b^2)\\\text{We get}\\a = x\\b = 9\\x^3-9^3 = (x-9)(x^2 + 9x + 81)[/tex]
Thus, it can be expressed as a difference of cubes.
4.
[tex](x - 10)(x^2 - 10x + 100)[/tex]
It is not a sum or difference of cubes because it does not satisfies the identity.
Answer:
C
Step-by-step explanation:
USA Today reports that about 25% of all prison parolees become repeat offenders. Alice is a social worker whose job is to counsel people on parole. Let us say success means a person does not become a repeat offender. Alice has been given a group of four parolees.(a) Find the probability P(r) of r successes ranging from 0 to 4. (Round your answers to three decimal places.)P(0) =P(1) =P(2) =P(3) =P(4) =(c) What is the expected number of parolees in Alice's group who will not be repeat offenders? (Round your answer to two decimal places.)What is the standard deviation? (Round your answer to two decimal places.)(d) How large a group should Alice counsel to be about 98% sure that three or more parolees will not become repeat offenders?
Answer:
a) [tex]P(X=0)=(4C0)(0.75)^0 (1-0.75)^{4-0}=0.0039[/tex]
[tex]P(X=1)=(4C1)(0.75)^1 (1-0.75)^{4-1}=0.0469[/tex]
[tex]P(X=2)=(4C2)(0.75)^2 (1-0.75)^{4-2}=0.211[/tex]
[tex]P(X=3)=(4C3)(0.75)^2 (1-0.75)^{4-3}=0.422[/tex]
[tex]P(X=4)=(4C4)(0.75)^2 (1-0.75)^{4-4}=0.316[/tex]
b) [tex] E(X) = np = 4*0.75=3[/tex]
c) [tex] Sd(X) =\sqrt{np(1-p)}=\sqrt{4*0.75*(1-0.75)}=0.866[/tex]
d) [tex] P(X \geq 3) \geq 0.98[/tex]
And the dsitribution that satisfy this is [tex]X\sim Binom(n=9,p=0.75[/tex]
Step-by-step explanation:
Previous concepts
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
Let X the random variable of interest, on this case we now that:
[tex]X \sim Binom(n=4, p=1-0.25=0.75)[/tex]
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
Part a
[tex]P(X=0)=(4C0)(0.75)^0 (1-0.75)^{4-0}=0.0039[/tex]
[tex]P(X=1)=(4C1)(0.75)^1 (1-0.75)^{4-1}=0.0469[/tex]
[tex]P(X=2)=(4C2)(0.75)^2 (1-0.75)^{4-2}=0.211[/tex]
[tex]P(X=3)=(4C3)(0.75)^2 (1-0.75)^{4-3}=0.422[/tex]
[tex]P(X=4)=(4C4)(0.75)^2 (1-0.75)^{4-4}=0.316[/tex]
Part b
The expected value is givn by:
[tex] E(X) = np = 4*0.75=3[/tex]
Part c
For the standard deviation we have this:
[tex]Sd(X) =\sqrt{np(1-p)}=\sqrt{4*0.75*(1-0.75)}=0.866[/tex]
Part d
For this case the sample size needs to be higher or equal to 9. Since we need a value such that:
[tex] P(X \geq 3) \geq 0.98[/tex]
And the dsitribution that satisfy this is [tex]X\sim Binom(n=9,p=0.75[/tex]
We can verify this using the following code:
"=1-BINOM.DIST(3,9,0.75,TRUE)" and we got 0.99 and the condition is satisfied.
To find the probabilities of different numbers of successes ranging from 0 to 4 in a group of parolees, we can use the binomial probability formula. The expected number of parolees who will not be repeat offenders can be calculated by multiplying the probability of success by the total number of parolees. The standard deviation can also be calculated using a formula. To be about 98% sure that three or more parolees will not become repeat offenders, Alice should counsel a group size of at least 13 parolees.
Explanation:To find the probabilities of different numbers of successes, we can use the binomial probability formula. Given that the probability of success is 0.25 and the number of trials is 4, we can calculate the probabilities as follows:
P(0) = (1-0.25)^4 = 0.316P(1) = 4C1 * 0.25^1 * (1-0.25)^3 = 0.421P(2) = 4C2 * 0.25^2 * (1-0.25)^2 = 0.281P(3) = 4C3 * 0.25^3 * (1-0.25)^1 = 0.047P(4) = 4C4 * 0.25^4 * (1-0.25)^0 = 0.008The expected number of parolees who will not be repeat offenders can be calculated by multiplying the probability of success (0.25) by the total number of parolees (4), which gives an expected value of 1 parolee. The standard deviation can be calculated using the formula sqrt(n * p * (1-p)), where n is the number of trials and p is the probability of success. In this case, the standard deviation is sqrt(4 * 0.25 * (1-0.25)) ≈ 0.866.
To determine the size of the group that Alice should counsel to be about 98% sure that three or more parolees will not become repeat offenders, we can use the binomial cumulative distribution function. We need to find the smallest value of n (the number of trials) such that P(X ≥ 3) > 0.98, where X represents the number of successes. Solving this inequality, we find that Alice should counsel a group size of at least 13 parolees.
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An arithmetic sequence begins as follows: a1=13 a2=19 Which of the following gives the definition of its nth term?
Answer:
the nth term of the sequence is [tex]a_n=6n+7[/tex]
Step-by-step explanation:
Given : An arithmetic sequence begins as follows: [tex]a_1=13, a_2=19[/tex]
To find : Which of the following gives the definition of its nth term?
Solution :
The nth term of the A.P is [tex]a_n=a+(n-1)d[/tex]
The first term is [tex]a=a_1=13[/tex]
The common difference is [tex]d=a_2-a_1[/tex]
[tex]d=19-13=6[/tex]
Substitute in the formula,
[tex]a_n=13+(n-1)6[/tex]
[tex]a_n=13+6n-6[/tex]
[tex]a_n=6n+7[/tex]
Therefore, the nth term of the sequence is [tex]a_n=6n+7[/tex]
The nth term of the given arithmetic sequence is defined by the formula an = 13 + (n-1)*6. This is derived from the general formula for an arithmetic sequence and using the given first two terms.
Explanation:In this given problem, we have an arithmetic sequence. An arithmetic sequence is a list of numbers in which each term is obtained by adding a constant difference to the preceding term. For this particular sequence, the first term (a1) is 13, and the second term (a2) is 19. Hence, the common difference (d) between the terms is 19 - 13 = 6.
The general formula for the nth term (an) of an arithmetic sequence is given by an = a1 + (n-1)*d. In this situation, to represent any term in the series, the formula would be an = 13 + (n-1)*6. This formula can generate any term in the sequence, given the term number.
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A hardware store rent vacuum cleaners that customers for part or all of A day before returning. The Store charges a flat fee Plus an hourly rate. Write a linear function F for the total retail cost of a vacuum cleaner.
A linear function can be used to represent the total retail cost of renting a vacuum cleaner: F(x) = 31.50 + 32x.
Explanation:A linear function can be used to represent the total retail cost of renting a vacuum cleaner. Let's denote the fixed fee as $31.50 and the hourly rate as $32. The linear function F for the total retail cost of a vacuum cleaner can be written as:
F(x) = 31.50 + 32x
Where x represents the number of hours the vacuum cleaner is rented for and F(x) gives the total retail cost.
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The flat fee that the store charges is $14 and the cost for 7 hours is $56
A linear equation is on the form:
y = mx + b
where y, x are variables, m is the rate of change and b is the initial value of y.
let f for the total rental cost of a vacuum cleaner for x hours
Using the points (1, 20) and (3, 32) from the table:
f-f1=(f2-f1)/(x2-x1) (x-x1)
f-20=(32-20)/(3-1) (x-1)
f(x)=6x+14
The flat fee that the store charges is $14
The reasonable domain is 1 ≤ x ≤ 12
The cost for 7 hours is:
f(7) = 6(7) + 14 = 46
here is the complete question-
A hardware store rents vacuum cleaners that customers may use for part or all of a day before returning. The store charges a flat fee plus an hourly rate. Part A Write a linear function f for the totall rental cost of the vacuum cleaner. A. f(x)=6x+14 B. f(x)=3x+14 C, f(x)=3x+22 D. f(x)=6x+24 Part B What is a reasonable domain for the function? A. 14 B. 1 C. 0 D. 20
least to greatest
A-7.2*10^-4 B+8.1*10^-6 C=1.2*10^4 D=9.5*10^5
answers A)D
B) B
C)B
D) D
Step-by-step explanation:
9.5 × 10^(5) is greater than 1.2 × 10^(4) is greater than 8.1 × 10^(-6) is greater than -7.2×10^(-4)
D > C > B > A
To order the numbers from least to greatest, one must compare the exponents in scientific notation. The correct sequence from least to greatest is B, A, C, D.
The question asks to order the given numbers from least to greatest. The numbers are provided in scientific notation, which is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. To compare these numbers, we look at their power of 10, as this indicates the number's scale or size.
'A' is 7.2*10^-4 which is a small positive number.
'B' is 8.1*10^-6 which is even smaller, given the more negative exponent.
'C' is 1.2*10^4 which is a large positive number.
'D' is 9.5*10^5 which is larger than 'C', given the larger exponent.
Therefore, when ordered from least to greatest, the sequence is: B, A, C, D.