Choose the correct graph of the given system of equations.
y + 2x = −1
3y − x = 4
A.) graph of two lines that intersect at the point negative 1, 1, with text on graph that reads One Solution negative 1, 1
B.) graph of two lines that intersect at the point 1, 1, with text on graph that reads One Solution 1, 1
C.) graph of two parallel lines with positive slopes with text on the graph that reads No Solution
D.) graph of two lines on top of each other with text on graph that reads Infinitely Many Solutions
Answers
A.) graph of two lines that intersect at the point negative 1, 1, with text on graph that reads One Solution negative 1, 1
Step-by-step explanation:
The two simultaneous equation given are;
y+2x= -1
3y-x=4
multiply the first equation by 1 and the second equation by 2 to make the terms with x equal
y+2x = -1
6y-2x=8
-------------- ---add the terms with x to eliminate x
7y=7 -------divide both sides by 7 to remain with y
7y/7 =7/7
y= 1 -----use the value of y in the first equation
y+2x= -1
1+2x= -1
2x= -2
2x/2= -2/2
x= -1
The solution is (-1,1)
You can plot the equations on a graph tool as shown below to visualize the solution where the two linear equations intersect
See attached graph
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graph of two lines that intersect at the point negative 1, 1, with text on graph that reads One Solution negative 1, 1
Related Questions
What is the measure of a°?
Answers
Answer:
The correct answer is C. 129.
Step-by-step explanation:
Let's recall that the Inscribed Quadrilateral Theorem states that a quadrilateral is inscribed in a circle if and only if the opposite angles are supplementary.
In the case of the inscribed quadrilateral ABCD in the graph attached, we have that:
m∠C= 51°, therefore its supplementary angle, ∠A, should be the difference between 180° and m∠C.
m∠C= 180° - m∠A
Replacing with the real values:
51 = 180 - m∠A
m∠A = 129°
The correct answer is C. 129.
math question in the attached text
Answers
Answer:
A 1.44 cm cubed
Step-by-step explanation:
Just multiply all the numbers to get the area of the block
Answer:
A 1.44 cm cubed
Step-by-step explanation:
Ann increased the quantities of all the ingredients in a recipe by 60\%60%60, percent. She used 808080 grams (\text{g})(g)left parenthesis, start text, g, end text, right parenthesis of cheese. How much cheese did the recipe require?
Answers
Answer:
50 grams
Step-by-step explanation:
Let the amount of cheese required by the recipe be "x"
Ann increased 60% from original amount and then used up 80 grams. Thus:
Original, increased by 60%, became 80
This translated to algebraic equation would be:
x + 0.6x = 80
Note: 60% = 60/100 = 0.6
So we can solve the above equation for "x" and get our answer. Shown below:
[tex]x + 0.6x = 80\\1.6x=80\\x=\frac{80}{1.6}\\x=50[/tex]
Hence,
the recipe required 50 grams of cheese
Rich is attending a 4-year college. As a freshman, he was approved for a 10-year, federal unsubsidized student loan in the amount of $7,900 at 4.29%. He knows he has the option of beginning repayment of the loan in 4.5 years. He also knows that during this non-payment time, interest will accrue at 4.29%.
Answers
Rich will accrue approximately $1,518.45 in interest during the 4.5-year nonpayment period on his federal unsubsidized student loan.
To calculate the interest that Rich will accrue during the 4.5-year nonpayment period on his federal unsubsidized student loan, we can use the formula for simple interest:
Interest = Principal (loan amount) x Rate x Time
Where:
Principal (loan amount) = $7,900
Rate = 4.29% (0.0429 as a decimal)
Time = 4.5 years
Now, plug these values into the formula:
Interest = $7,900 x 0.0429 x 4.5
Interest = $7,900 x 0.19205
Interest ≈ $1,518.45
So, Rich will accrue approximately $1,518.45 in interest during the 4.5-year nonpayment period on his federal unsubsidized student loan.
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Rich has been approved for a federal unsubsidized student loan and wants to know the interest that will accrue during his 4.5 years as a freshman in college. Calculating the interest using the formula, the total accrued interest is $1,146.75.
Explanation:Rich is attending a 4-year college and has been approved for a 10-year federal unsubsidized student loan of $7,900 at an interest rate of 4.29%. During his 4 years as a freshman, he has the option to start repaying the loan after 4.5 years. However, interest will continue to accrue at a rate of 4.29% during this non-payment period.
To calculate the interest that will accrue during the 4.5 years, we can use the formula:
Interest Accrued = Principal x Interest Rate x Time
Substituting the values, we get:
Interest Accrued = $7,900 x 0.0429 x 4.5 = $1,146.75
Therefore, the interest that will accrue during the non-payment time is $1,146.75.
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The Davis family traveled 35 miles in 1/2 of an hour. If it is currently 2;00 pm and the family destination is 245 miles away at what time will they arrive there explain how you solved the problem
Answers
Answer:
5.30 PM
Step-by-step explanation:
time taken=(1/2)×(1/35)×245=7/2 hrs=3hrs 30 min.
so time =2+3.30=5.30 PM
Answer: 5:30pm
Step-by-step explanation:
Distance of family destination from the starting point = 245miles
Average speed = 35miles/1/2hour
Therefore 245 miles = 245/35 x 30mins.
7 x 30 = 210minutes.
Converted to hour by dividing by 60
210/60 = 3hours 30minutes.
Current local time at the point of commencement
Therefore, the arrival time at their destination = 14hours + 3hours 30mins.
= 17hours 30minutes. Convert to local time
17hours 30minutes = 12hours
= 5:30 pm in the evening
They have succeeded in spending 3hours 30minutes on the road. The final answer = 5:30pm.
If you have a lowest score of 21 and a range of 47, your highest score will be:________
Answers
Answer: 68
Step-by-step explanation:
range=47 lowest score = 21 let highest score =x
Range=highest score - lowest score
47 = x - 21
x = 47 + 21
x = 68
Therefore the highest score is 68
The highest score is 68.
Range = Highest Score - Lowest Score
In this case,
the lowest score is 21 and
the range is 47.
So, we can rearrange the formula to solve for the highest score:
Highest Score = Range + Lowest Score
Substituting in the given values:
Highest Score = 47 + 21
Thus, the highest score will be 68.
A 20 kg sphere is at the origin and a 10 kg sphere is at x = 20 cm. At what position on the x-axis could you place a small mass such that the net gravitational force on it due to the spheres is zero?
Answers
Step-by-step explanation:
Let the small mass be m and position on x axis be y.
A 20 kg sphere is at the origin.
Distance to 20 kg mass = y
A 10 kg sphere is at x = 20 cm
Distance to 10 kg sphere = 20 - y
We have forces between them are equal
We have gravitational force
[tex]F=\frac{GMm}{r^2}[/tex]
Where G = 6.67 x 10⁻¹¹ N m²/kg²
M = Mass of body 1
M = Mass of body 2
r = Distance between them
Here we have
[tex]\frac{G\times 20\times m}{y^2}=\frac{G\times 10\times m}{(20-y)^2}\\\\800-80y+2y^2=y^2\\\\y^2-80y+800=0\\\\y=68.28cm\texttt{ or }y=11.72cm[/tex]
So the small mass should be placed at x = 11.72 cm or x = 68.28 cm
The positions on the x-axis obtained using quadratic equation such that net gravitational force is zero are 68.28 and 11.72 cm respectively.
Recall the force of universal gravitational attraction :
[tex] \frac{Gm_{1}m_{2}}{r^{2}}[/tex]Sphere 1 :
Mass = 20 kg Distance of 20kg sphere = dSphere 2 :
Mass = 10kgDistance of 10kg sphere = 20 - dGravitational force ; sphere 1 :
[tex] \frac{G \times 20 \times m_{2}}{d^{2}}[/tex] --(1)Gravitational force ; sphere 2 :
[tex] \frac{G \times 10 \times m_{2}}{(20 - d)^{2}}[/tex] - - (2)Equate (1) and (2)
[tex] \frac{G \times 20 \times m_{2}}{d^{2}} = \frac{G \times 10 \times m_{2}}{(20 - d)^{2}} [/tex]
[tex] 20(20 - d)^{2} = 10 \times d^{2} [tex]
20(400 - 40d + d²) = 10d²
8000 - 800d + 20d² = 10d²
10d² - 800d + 8000 = 0
Divide through by 10
d² - 80d + 800 = 0
Using a quadratic equation solver :
d = 68.28 cm or d = 11.72 cm
The possible positions are 68.28 cm and 11.72 cm
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Find the m∠p.
pls and thanks <3
Answers
Answer:
36°
Step-by-step explanation:
Like your other question, the angles of the triangle must add up to 180. The tangent line is perpendicular to the center, so the angle must be 90°.
90° + 54° + 36° = 180°
Sketch the graph of the given function. Then state the function’s domain and range. f(x)= 1/2(5)^x+3
Answers
Answer:
Graph of the following function is attached with the answer.
Domain : ( - ∞ , + ∞ )
Range : ( 3 , + ∞ )
Step-by-step explanation:
[tex]f(x)=\frac{1}{2}x^{2}+3[/tex]
Domain of any quadratic equation is from negative infinity to positive infinity under no restrictions.
So, Domain : ( - ∞ , + ∞ )
The Range of any function can be calculated easily if there is just one term with variable. The method to find Range by that method is explained with the example as follows:
Range of x : ( - ∞ , + ∞ )Range of [tex]\textrm{x}^{2}[/tex] : [ 0 , + ∞ ) as every square number is more than or equal to zero.Range of [tex]\frac{1}{2}\textrm{x}^{2}[/tex] : [ 0 , + ∞ ) as 0/2 = 0 and ∞/2 = ∞.Range of [tex]\frac{1}{2}\textrm{x}^{2}+3[/tex] : [ 3 , + ∞ ) as 0 + 3 = 3 and ∞ + 3 = ∞.Therefore the Range of [tex]\mathbf{f(x)\boldsymbol=\frac{1}{2}x^{2}\boldsymbol+3}[/tex] is [ 3 , + ∞ )
(NOTE : [a,b] means all the numbers between 'a' and 'b' including 'a' and 'b'.
(a,b) means all the numbers between 'a' and 'b' excluding 'a' and 'b'.
(a,b] means all the numbers between 'a' and 'b' including only 'b' not 'a'.
[a,b) means all the numbers between 'a' and 'b' including only 'a' not 'b'.
{a,b} means only 'a' and 'b'.
{a,b] or (a,b} doesn't mean anything. )
Which expression gives the area of the RED rectangle.
A
(A + B)(C + D)
B
(A + B)(C - D)
C
(A + B)(C + D) - C(A + B) - BD
D
(A + B)(C + D) - D(A + B) - BD
Answers
The expression which gives the area of the RED rectangle is (A+B)(C+D)-C(A+B)-BD. option C is correct.
The area of rectangle is obtained by multiplying the length and width.
The length of the rectangle is C+D
The width of the rectangle is A+B.
Now the complete area of rectangle :
Area = (A+B)(C+D)
Now to find area of rectangle which is red we have to subtract the red rectangle which are in blue:
The area of left side rectangle which is blue:
Area =C(A+B)
The area of rectangle below red rectangle:
Area =BD
So the area of red rectangle : (A+B)(C+D)-C(A+B)-BD.
Hence, option C is correct, the expression which gives the area of the RED rectangle is (A+B)(C+D)-C(A+B)-BD.
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What additional information will allow you to prove the triangles congruent by the HL Theorem?A. A = EB. bce= 90C. ac = dcD. ac=bd
Answers
Answer:
C) AC=DC
Step-by-step explanation:
In the figure of question attached below:
In Δ ACB and Δ DCE, AB and DE are hypotenuse respectively.
According to HL theorem:
"If hypotenuse and one leg of right angle triangle is congruent to Hypotenuse and one leg of other right angle triangle then triangles are congruent"
According to above statement if AB and DE are hypotenuse of Δ ACB and Δ DCE respectively then either
AC = DC (leg)
BC = EC (leg)
In order to prove congruence of triangles using HL theorem, AC must be equal to DC.
So option C is correct.
Answer:
C ) AC=DC
Step-by-step explanation:
edge 2021
Susan has 5 more pennies than dimes, three times more nickels than dimes and 4 less quarters than pennies. If Susan has 7 dimes, what is the total value of her money?
Answers
To solve this, let’s first write equations for each of the coins listed. Let’s start with pennies.
We know that Susan has 3 more pennies than dimes. This means that pennies is equal to dimes, plus three. This would turn into this equation:
Pennies: d+3
Now let’s move onto nickels. The question says that Susan has three times more nickels than dimes. This means that nickels is equal to dimes, times three. This would be our equation:
Nickels: n=3d
Finally, let’s write the equation for quarters. It says that Susan has four less quarters than pennies. This means that quarters are equal to pennies minus three. This would be the equation:
Quarters: q=p-4
Now time to solve. We need to solve the amount of...
Pennies: ?
Nickels: ?
Dimes: ?
Quarters: ?
The question tells us that Susan has 7 dimes, so let’s fill that in now.
Pennies: ?
Nickels: ?
Dimes: 7
Quarters: ?
We can use the dime information to solve for the other coins, using our equation. Let’s go in order and start with pennies.
Our equation for pennies is: p=d+3. We know that d is 7, so let’s fill this in and solve.
p=d+3
p=7+3
p=10
Susan has 10 pennies. Let’s fill this into the information we have.
Pennies: 10
Nickels: ?
Dimes: 7
Quarters: ?
Now let’s do nickels. Our equation says n=3d. Let’s substitute the value for d and solve
n=3d
n=3(7)
n=21
Susan has 21 nickels. Let’s fill that into the information we have.
Pennies: 10
Nickels: 21
Dimes: 7
Quarters: ?
Let’s finally do quarters. The equation we have written for quarters is: q=p-4. Let’s fill in the value of p and solve
q=p-4
q=10-4
q=6
Susan has 6 quarters. This is our final information on the coins:
Pennies: 10
Nickels: 21
Dimes: 7
Quarters: 6
Let’s now calculate how much money she has altogether. We will do this by multiplying how many coins she has by the value of each coin.
Pennies: 10 • .01= .10
Nickels: 21•.05=1.05
Dimes: 7•.10=.70
Quarters: 6•.25=1.5
Now that we know that, let’s add these all together. This will tell us the final answer of how much is the value of her money.
.10+1.05+.70+1.5= $3.35
That’s your answer! Hope this helps comment below for more questions :)
2. Jenna Ford purchased a compact stereo that regularly sells for $229.99 at a markdown rate of 53%. In California, where Jenna lives, the sales tax rate on this item is 7%. Find the final price of the item.
Answers
Answer:
$115.67
Step-by-step explanation:
given that the markdown rate is 53%, that means that the markdown is 53% of the original price, or,
markdown = 53% of $229.99 = 0.53 x 229.99 = $121.89
sale price before tax = Original Price - Markdown
= 229.99 - 121.89
= $108.10
sales tax = 7% of sale price before tax
= 0.07 x 108.10
= $7.57
Final price = price before tax + tax
= 108.1 + 7.57
= $115.67
State if the triangles in each pair are similar. If so, State how you know they are similar and complete the similarity statement.
Answers
Answer:
Step-by-step explanation:
Looking at both triangles, angle P in triangle PQR = 38 degrees. Angle N in triangle LMN = 38 degrees. Both angles are equal.
Side PQ in triangle PQR = 16
Side MN in triangle LMN = 8
Therefore,
PQ/MN = 16/8 = 2
Side PR in triangle PQR = 14
Side LN in triangle LMN = 7
Therefore,
PR/LN = 14/7 = 2
Therefore, triangle PQR is similar to
triangle LMN because
1) the length of PQ is proportional to the length of MN.
2) the length of PR is proportional to the length of LN
3) angle P = angle N
4) Therefore, QR is also proportional to ML
Therefore,
PQ/MN = PR/LN = QR/ML = 2
21. What is the exact volume of a sphere whose diameter is 14 cm? Show your work.
Answers
Answer:
The exact volume of a sphere is [tex]V=1437cm^3[/tex]
Step-by-step explanation:
Given that the diameter of a sphere is 14cm.
That is d=14
First find the radius of the sphere with diameter
[tex]r=\frac{d}{2}[/tex]
[tex]r=\frac{14}{2}[/tex]
Therefore r=7cm
Now to find the volume of the sphere:
[tex]V=\frac{4}{3}\pi r^3cm^3[/tex]
[tex]V=\frac{4}{3}\frac{22}{7}(7)^3[/tex] (here[tex]\pi=\frac{22}{7}[/tex] and r=7)
[tex]V=\frac{4}{3}(22)(7)^2[/tex]
[tex]V=\frac{4}{3}(22)(49)[/tex]
[tex]V=\frac{4}{3}(1078)[/tex]
[tex]V=\frac{4312}{3}[/tex]
[tex]V=1437.33cm^3[/tex]
Therefore the volume of the given sphere is [tex]V=1437.33cm^3[/tex]
The exact volume of a sphere is [tex]V=1437cm^3[/tex]
Jake, a varsity swimmer, is training for his upcoming season and needs to accelerate his post-exercise recovery rates. He trains intensely for 2 hours, 5 days a week. If he weighs 155-lb (70.45 kg), which of the following nutritional strategies would BEST replenish his glycogen stores?
Answers
Answer:
high glycemic carb with some protein
Step-by-step explanation:
Food like white bread made from dextrose supplement and chocolate milk that contains some protein for muscle repair.
Due to the spongelike characteristic of the muscle would make it soak up glucose from high glycemic carb.
The scale model of a rectangular garden is 1.5 ft by 4 ft. The scale model is enlarged by a scale factor of 7 to create the actual garden. What is the area of the actual garden
Answers
Answer:
The Area of the actual garden is 294 square feet.
Step-by-step explanation:
The scale model of a rectangular garden is 1.5 ft by 4 ft.
Length of scale model=1.5 ft
Breath of scale model=4 ft
The scale model is enlarged by a scale factor of 7 to create the actual garden.
It means that the dimension of the garden are multiplied with the scale factor to find the actual dimension.
Hence,
Length of actual garden=[tex]1.5\times 7= 10.5\ ft[/tex]
Breath of actual garden=[tex]4\times 7 = 28\ ft[/tex]
Now Area of garden can be calculated by multiplying length and breadth.
Framing the equation we get;
Area of actual garden=[tex]10.5\ ft \times 28\ ft = 294\ ft^2[/tex]
Hence, area of the actual garden is 294 square feet.
Answer:294 took test
On July 31, Oscar checked out of the Sandy Beach hotel after spending four nights. The cost of the room was $76.90 per night. He wrote a check to pay for his four-night stay. What was the total amount of his check?
Answers
Answer:
$307.60
Step-by-step explanation:
multiple $76.90 by the 4 nights he stayed. $76.90×4
Answer:
76.00
Step-by-step explanation:
£980 is divided between Caroline, Sarah & Gavyn so that Caroline gets twice as much as Sarah, and Sarah gets three times as much as Gavyn. How much does Sarah get?
Answers
Sarah has received £ 294
Solution:
Given that £980 is divided between Caroline, Sarah & Gavyn
Let "c" be the amount received by caroline
Let "s" be the amount received by sarah
Let "g" be the amount received by gavyn
Caroline gets twice as much as Sarah
amount received by caroline = twice as much as Sarah
amount received by caroline = 2(amount received by sarah)
c = 2s ---- eqn 1
Sarah gets three times as much as Gavyn
amount received by sarah = three times as much as Gavyn
amount received by sarah = 3(amount received by gavyn)
s = 3g ------- eqn 2
Given that total amount is 980
c + s + g = 980 --- eqn 3
Let us solve eqn 1, 2, 3 to get values of "c" "s" "g"
From eqn 2,
[tex]g = \frac{s}{3}[/tex] --- eqn 4
Substitute eqn 1 and eqn 4 in eqn 3
[tex]2s + s + \frac{s}{3} = 980\\\\\frac{6s + 3s + s}{3} = 980\\\\6s + 3s + s = 980 \times 3\\\\10s = 2940\\\\s = 294[/tex]
Thus sarah has received £ 294
Final answer:
To solve for the amount Sarah receives, set up ratios based on the information provided and solve the resulting equation. The total amount divided among them is £980, which when divided by the total parts (10) gives Gavyn's share as £98. Sarah's share is three times Gavyn's share, resulting in £294.
Explanation:
The problem described is a classic division in ratio mathematics question where £980 is being divided among Caroline, Sarah, and Gavyn following certain rules.
According to the problem, Caroline receives twice the amount that Sarah receives and Sarah receives triple the amount that Gavyn receives.
We can set up the following ratios: C = 2S and S = 3G, where C stands for Caroline's amount, S for Sarah's, and G for Gavyn's. If we denote Gavyn's amount as G, then Sarah's amount is 3G and Caroline's is 2 × 3G which is 6G.
To find the value of G, we can write the equation G + 3G + 6G = £980 or 10G = £980. Solving for G gives us G = £98.
Therefore, Sarah, receiving three times as much as Gavyn, gets 3 × £98 = £294.
Your grandmother always has a jar of cookies on her counter. One day while you are visiting, you eat 555 cookies from the jar. In the equation below, ccc is the number of cookies remaining in the jar and bbb is the number of cookies in the jar before your visit.
Answers
Answer:
For 1 How does grandma have so many cookies??? For 2 if i eat 555 cookies then imma have diabetes. and for 3 there really is no way to find this out. The only way to find out how many cookies there is now it to see how much cookies there were before and you cant do that without seeing how much is in there now. Its a catch 22.
Step-by-step explanation:
Can someone please write the equation of hyperbola with vertices (0, -5) and (0, 5) and co-vertices (-2, 0) and (2, 0)?
Thank you so much!
Answers
Answer:
y^2/25 -x^2/4 = 1
Step-by-step explanation:
The equation can be written as ...
y^2/a^2 -x^2/b^2 = 1
where the vertices are at (0, ±a) and the co-vertices are at (±b, 0). Filling in the given values (a=5, b=2) gives the equation shown above.
write a verbal phrase to describe f > -4
Answers
I would say it as "the letter F is greater than negative four."
Three pounds of dried cherries cost $15.90, 5 pounds of dried cherries cost $26.50, and 9 pounds of dried cherries cost $47.70. Which equation gives the total cost y of x pounds of dried cherries?
Answers
Final answer:
The equation that gives the total cost of x pounds of dried cherries is y = (10.60/2)x.
Explanation:
Let's create a table to represent the given information:
Pounds of Dried Cherries (x)Total Cost (y)315.90526.50947.70
To find an equation that gives the total cost of x pounds of dried cherries, we need to find a pattern in the data. From the table, we can see that as the pounds of dried cherries increase, the total cost also increases. This suggests a linear relationship between pounds and cost.
Now, let's analyze the changes in cost for each additional pound of dried cherries:
From 3 to 5 pounds: The cost increased by $26.50 - $15.90 = $10.60.From 5 to 9 pounds: The cost increased by $47.70 - $26.50 = $21.20.Based on these changes, the cost increased by $10.60 for every 2 additional pounds of dried cherries. Therefore, the equation that gives the total cost y of x pounds of dried cherries is:
y = (10.60/2)x
In an arcade game, a 0.13 kg disk is shot across a frictionless horizontal surface by being compressed against a spring and then released. The spring has a spring constant of 242 N/m and is compressed from its equilibrium position by 5.2 cm. What is the magnitude of the spring force on the disk at the moment it is released?
Answers
Answer:
12.584 N
Step-by-step explanation:
To solve this problem you need to use Hokke's law, this is a physics lar which states the force (F) you need to compress a spring for some distance (x) can be easily calculated with the equation F=kx. The constant k (the stiffness of the spring) is the value of 242 N/m and x is the distance 5.2 cm = 0.052 m. So if you multiply 242 N/m by 0.052 m you will obtain 12.584 N, which is the necessary force to compress the spring 5.2 cm. The mass of the spring is a nonrelevant data in this problem.
The measures of one acute angle in a right triangle is four times the measure of the other acute
angle. Write and solve a system of equations to find the measures of the acute angles.
Help with this exercise
Answers
[tex]A_2[/tex] = 18°
[tex]A_3[/tex] = 4(18°) = 72°
Step-by-step explanation:Given:
One angle in the triangle is 90°One angle that isn't 90° is 4 times larger than another angle that isn't 90°Angles:
[tex]A_1[/tex] = 90°
[tex]A_2[/tex] = x
[tex]A_3[/tex] = 4x
Solution Pathway:
Under the rules for any triangle, a triangle's interior angles must add up to 180°. Using this, we can set up the equation:
sum of the interior angles = 180°90° + x + 4x = 180°Now let's solve for x.
90 +x + 4x = 18090 + 5x = 1805x = 90x = 18°Now that we know x is 18°, lets plug this value into the two unknown acute angles.
[tex]A_2[/tex] = 18°[tex]A_3[/tex] = 4(18°) = 72°Answer:
72 degrees and 18 degrees.
Step-by-step explanation:
If the 2 angles are x and y, we have the system:
x + y = 90 (as it is a right triangle)
x = 4y (given).
Substitute x =- 4y in the first equation:
4y + y = 90
5y = 90
y = 18.
So x + 18 = 90
x = 90 - 18
x = 72.
Here are the records of two different sequences (A,B ) of a coin tossed eight times. A: T H H H H H H H B: H T T H T H H T If you know for sure that the coin is fair, are these two sequences equally probable outcomes or, if they are not, which sequences is more probable than the other?"
Answers
Answer: Sequence B is more probable than A.
Step-by-step explanation:
This two sequences are not equally probable. Sequence B is more probable than A due to the equal chances of getting head (H) and a tail (T). The probability of getting a head is equal to the probability of getting a tail which is 4/8 i.e 0.5
The sequence A is less probable because the head(H) occur more than tail (T). The probability of head occurring is almost a sure event i.e 1 which is not feasible.
Let A be a set with a partial order R. For each a∈A, let Sa= {x∈A: xRa}. Let F={Sa: a∈A}. Then F is a subset of P(A) and thus may be partially ordered by ⊆, inclusion.
a) Show that if aRb, then Sa ⊆ Sb.
b) Show that if Sa ⊆ Sa, then aRb.
c) Show that if B⊆A, and x is the least upper bound for B, then Sx is the least upper bound for {Sb:b∈B}
Answers
Answer:
See proofs below
Step-by-step explanation:
a) Suppose that aRb. Let y∈Sa , then y∈A and yRa. We have that yRa and aRb. Since R is a partial order, R is a transitive relation, therefore yRa and aRb imply that yRb. Now, y∈A and yRb, thus y∈Sb. This reasoning applies for all y∈Sa that is, for all y∈Sa, y∈Sb, threrefore Sa⊆Sb.
b) Suppose that Sa⊆Sb. Since R is a partial order, R is a reflexive relation then aRa. Thus, a∈Sa. The inclusion Sa⊆Sb implies that a∈Sb, then aRb.
c) Denote this set by S={Sb:b∈B}. We will prove that supS=Sx with x=supB.
First, Sx is an upper bound of S: let Sb∈S. Then b∈B and since x=supB, x is an upper bound of B, then bRx. Then b∈Sx. Now, for all y∈Sb, yRb and bRx, then by transitivity yRx, thus y∈Sx. Therefore Sb⊆Sx for all Sb∈S, which means that Sx is an upper bound of S (remember that the order between sets is inclusion).
Now, let's prove that Sx is the least upper bound of S. Let Sc⊆A be another upper bound of S (in the set F). We will prove that Sx⊆T.
Because Sc is an upper bound, Sb⊆Sc for all b∈B. Thus, if y∈Sb for some b, then y∈Sc. That is, if yRb then yRc. In particular, bRb then bRc for all b∈B. Thus c is a upper bound of B. Byt x=supB, then xRc. Now, for all z∈Sx, zRx and xRc, which again, by transitivity, implies that z∈Sc. Therefore Sx⊆Sc and Sx=sup S.
In a partial order set, if aRb, then Sa ⊆ Sb. If Sa ⊆ Sb, then aRb. If B ⊆ A and x is the least upper bound for B, then Sx is the least upper bound for {Sb : b ∈ B}.
Explanation:a) To show that if aRb, then Sa ⊆ Sb, we need to prove that if x ∈ Sa, then x ∈ Sb. Since x ∈ Sa, that means xRa holds. And since aRb, we can conclude that xRb holds as well. Therefore, x ∈ Sb, which implies that Sa ⊆ Sb.
b) To show that if Sa ⊆ Sb, then aRb, we need to prove that if aRb does not hold, then Sa ⊆ Sb does not hold. If aRb does not hold, it means that b is not in Sa. However, if Sa ⊆ Sb, it implies that every element in Sa is also in Sb. Hence, we arrive at a contradiction, which proves that if Sa ⊆ Sb, then aRb.
c) To show that if B ⊆ A and x is the least upper bound for B, then Sx is the least upper bound for {Sb : b ∈ B}, we need to prove that Sx is an upper bound for {Sb : b ∈ B} and that it is the least upper bound. Since x is the least upper bound for B, this means that every element of B is contained in Sx. Therefore, Sx is an upper bound for {Sb : b ∈ B}. Additionally, if there exists an upper bound U for {Sb : b ∈ B}, then every element of {Sb : b ∈ B} must be contained in U. Since Sb is a subset of Sa for every b ∈ B, it follows that every element of Sa is contained in U. Therefore, Sa ⊆ U for every a ∈ A. In particular, Sx ⊆ U, which proves that Sx is the least upper bound for {Sb : b ∈ B}.
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The function f(x) = 6x + 8 represents the distance run by a cheetah in miles. The function g(x) = x − 2 represents the time the cheetah ran in hours. Solve (f/g)(3), and interpret the answer.
a. 26; the cheetah's rate in miles per hour
b. 26; the number of cheetahs
c. 1/26 ; the cheetah's rate in miles per hour
d. 1/26 ; the number of cheetahs
Answers
Option a. 26; the cheetah's rate in miles per hour is the correct answer.
Step-by-step explanation:
Given
[tex]f(x) = 6x+8\\g(x) = x-2[/tex]
As the function f is in miles and function g is is hours, and we are dividing the function f by function g so the new unit will be:
miles/hour = miles per hour
Now
[tex]\frac{f}{g}(x)= \frac{f(x)}{g(x)}\\\frac{f}{g}(x) = \frac{6x+8}{x-2}[/tex]
We have to find (f/g)(3) so putting 3 in place of x
[tex]\frac{f}{g}(x) = \frac{6(3)+8}{-2}\\= \frac{18+8}{1}\\= 26[/tex]
Hence,
Option a. 26; the cheetah's rate in miles per hour is the correct answer.
Keywords: Functions, function operations
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Match the square root with its perfect square: 1 . √4 1 2 . √144 2 3 . √9 3 4 . √121 4 5 . √64 5 6 . √169 6 7 . √100 7 8 . √25 8 9 . √1 9 10 . √36 10 11 . √81 11 12 . √16 12 13 . √49 13
Answers
Answer:
√1 1
√4 2
√9 3
√16 4
√25 5
√36 6
√49 7
√64 8
√81 9
√100 10
√121 11
√144 12
√169 13
Step-by-step explanation:
1. = 49
2. = 169
3. = 81
4. = 100
5. = 441
6. = 36
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A chain letter works as follows: One person sends a copy of the letter to five friends, each of whom sends a copy to five friends, each of whom sends a copy to five friends, and so forth. How many people will have received copies of the let- ter after the twentieth repetition of this process, assuming no person receives more than one copy?
Answers
Answer:
The number of people that received copies of the letter at the twentieth stage is 9.537 × 10¹³ .
Step-by-step explanation:
Using the discrete model,
a_k = r a_(k-1) for all integers k ≥ 1 and a₀ = a
then,
aₙ = a rⁿ for all integers n ≥ 0
Let a_k be equal to the number of people who receive a copy of the chain letter at a stage k.
Initially, one person has the chain letter (which the person will send to five other people at stage 1). Thus,
a = a₀ = 1
The people who received he chain letter at stage (k - 1), will send a letter to five people at stage k and thus per person at stage (k - 1), five people will receive the letter. Therefore,
a_k = 5 a_(k - 1)
Thus,
aₙ = a rⁿ = 1 · 5ⁿ = 5ⁿ
The number of people that received copies of the letter at the twentieth stage is
a₂₀ = (5)²⁰ = 9.537 × 10¹³ copies