Answer:
Step-by-step explanation:
I believe the answer is x² + y² = 9²
The amount of turn between two straight lines that have a common end point is called __________.
Answer:
Angle
Step-by-step explanation:
In the image I added you can observe that the common endpoint is called a vertex, and the "turn" or aperture formed between the two straight lines (or arms) that converge in the vertex, is the angle and it can be measured on degrees.
I hope you find this information usefun and interesting! Good luck!
Final answer:
The amount of turn between two straight lines that have a common end point is referred to as an angle. This concept is central to the study of geometry and aids in measuring the difference in direction where two lines meet. Angles are typically measured in degrees as a part of spherical trigonometry and other geometric calculations.
Explanation:
The amount of turn between two straight lines that have a common end point is called an angle. An angle measures the difference in direction between two lines that extend from the same point. The concept of angle is found in the principles of geometry, where it is foundational to understanding shapes and their properties.
In more mathematical terms, the angle between two intersecting lines can also be seen as the distance between their poles when conceptualized on a sphere, a concept from advanced geometry and spherical trigonometry. The measurement of this angle is generally given in degrees, where a full circle is 360 degrees, and this system of measurement allows us to apply the angle measure to various geometric constructions and theorems, such as determining the sum of the angles of a triangle.
Using the dot plots provided, what is the mean absolute deviation of the height of basketball players, rounded to the nearest tenth?
Answer:
[tex] MAD= \frac{1}{14} 27.286 = 1.949 \approx 2.0[/tex]
Step-by-step explanation:
Previous concepts
The mean absolute deviation or MAD "is the average of the absolute deviations or the positive difference of the given data and that certain value (generally central values)". And is given by this formula:
[tex] MAD= \frac{1}{n} \sum_{i=1}^n |x_i -\bar X|[/tex]
Solution to the problem
Assuming the info from the picture. So then the data is this one:
66,69,70,70,71,72,72,72,73,73,74,75,75,75
So the first step is find the mean for the dataset with the following formula:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
And if we replace the values we got:
[tex]\bar X = \frac{66+69+70+70+71+72+72+72+73+73+74+75+75+75
}{14}=71.929[/tex]
And now we need to subtract for each value the mean like this:
Data [tex] X_i -\bar X[/tex]
66 5.929
69 2.929
70 1.929
70 1.929
71 0.929
72 0.0714
72 0.0714
72 0.0714
73 1.0714
73 1.0714
74 2.0714
75 3.0714
75 3.0714
75 3.0714
Now we need to add the deviations and divide by the the number of data values and we got:
[tex] MAD= \frac{1}{14} 27.286 = 1.949 \approx 2.0[/tex]
1. Frank Choi bought a rechargeable lantern that regularly sells for $29.99. The markdown rate was 30% and there is no sales tax on the item. Find the final price of the item.
Answer:20.993
Step-by-step explanation:
The real price "r" is
100% and =29.99
The price after discount "d" is
100% - 30% = 70% and =?
So:
"r" is 100% =29.99
"d" is 70% =?
Do a cross multiplication so :
"d"= (29.99 * 70) / 100 = 20.993
The final price of the item is 20.993.
Frank Choi bought a rechargeable lantern that regularly sells for $29.99.
The markdown rate was 30% and there is no sales tax on the item.
The real price "r" is
100% =29.99
The markdown rate was 30%
29.99 * (30/100)
8.997
The price after discount
29.99 - 8.997 = 20.993.
Therefore, the final price of the item is 20.993.
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On a farm A 4/5 of the number of sheep are equal to 1/2 of the number of sheep on farm B. The total number of sheep on farm A and farm B is 845. How many sheep are there on farm B
Answer:
There are 520 sheep on Farm B.
Step-by-step explanation:
Let the Number of sheep on farm B be x
Given:
On a farm A 4/5 of the number of sheep are equal to 1/2 of the number of sheep on farm B.
It means Number of sheep on farm A multiplied by 4/5 is equal to 1/2 multiplied by Number of sheep on farm B.
framing the equation we get;
[tex]\frac{4}{5}\times \textrm{Number of sheep on Farm A} = \frac{1}{2} \times x[/tex]
Number of sheep on Farm A = [tex]\frac{x}{2}\times\frac{5}{4} = \frac{5x}{8}[/tex]
Now Given:
Total Number of sheep =845.
We know that Total Number of sheep is equal to sum of sheep at farm A and sheep at farm B.
Framing equation we get;
[tex]\frac{5x}{8}+x=845[/tex]
Taking L.C.M we get;
[tex]\frac{5x\times 1}{8\times1}+\frac{x\times8}{8} =845\\\\\frac{5x}{8}+\frac{x}{8} =845\\\\\frac{5x+8x}{8}=845\\\\\frac{13x}{8}= 845\\\\13x=845\times8\\\\x=\frac{845\times8}{13} = 520[/tex]
Number of sheep on Farm B = 520 sheep
Number of sheep on Farm A = [tex]\frac{5}{8}\times 520 = 325\ sheep[/tex]
Hence There are 520 sheep on Farm B.
Write an equation of the line containing the given point and perpendicular to the given line.
(7, -4); 9x+7y=4
please try to help i'm really struggling with this one:)
Answer:
Step-by-step explanation:
The equation of a straight line can be represented in the slope intercept form as
y = mx + c
Where
m = slope = (change in the value of y on the vertical axis) / (change in the value of x on the horizontal axis)
The equation of the given line is
9x+7y=4
7y = 4 - 9x = -9x + 4
y = -9x/7 + 4/7
Comparing with the slope intercept form, slope = -9/7
If the line passing through the given point is perpendicular to the given line, it means that its slope is the negative reciprocal of the slope of the given line.
Therefore, the slope of the line passing through (7,-4) is 7/9
To determine the intercept, we would substitute m = 7/9, x = 7 and y = - 4 into y = mx + c. It becomes
- 4 = 7/9×7 + c = 49/9 + c
c = - 4 - 49/9
c = - 85/9
The equation becomes
y = 7x/9 - 85/9
Find the exponential function that satisfies the given conditions: Initial value = 64, decreasing at a rate of 0.5% per week
f(t) = 0.5 ⋅ 0.36t
f(t) = 64 ⋅ 1.005t
f(t) = 64 ⋅ 0.995t
f(t) = 64 ⋅ 1.5t
Answer:
Step-by-step explanation:
An exponential function is of the form
[tex]y=ab^x[/tex]
where a is the initial value and b is the growth/decay rate. Our initial value is 64. That's easy to plug in. It goes in for a. So the first choice is out. Considering b now...
If the rate is decreasing at .5% per week, this means it still retains a rate of
100% - .5% = 99.5%
which is .995 in decimal form.
b is a rate of decay when it is greater than 0 but less than 1; b is a growth rate when it is greater than 1. .995 is less than 1 so it is a rate of decay. The exponential function is, in terms of t,
[tex]f(t) = 64(.995)^t[/tex]
Answer:
f(t) = 64 ⋅ 0.995t
Step-by-step explanation:
Add me on S n a p c h a t :) yofav_tai
For her birthday, Kendra received a gift card in the mail from her grandparents. The gift card was worth $50. Kendra thought, since she had just received other gifts for her birthday, she would wait a while to spend the gift card. Unfortunately, Kendra forgot all about the gift card, and the card loses $5 every month after its date of purchase if it has not been used.
If Kendra forgot the card for three months, she lost $______.
a) 9 b) 10 c) 15 d) 12
Answer:
She lost $15 on her 50 dollar credit card.
Step-by-step explanation:
Kendra received a gift card in the mail from her grandparents. The gift card was worth $50.
Kendra forgot all about the gift card, and the card loses $5 every month after its date of purchase if it has not been used.
Kendra loses $5 each month,
She forget the card for 3 months.
So, Multiply the $5 by 3 months
$5×3=$15
Hence, she lost $15.
David made a scale drawing of a mural he is going to paint. He used a scale of 1 cm=4ft. If the actual mural will be 16 feet long, how long is the mural in David's drawing
Answer:
4 cm
Step-by-step explanation:
we know that
The scale drawing is
[tex]\frac{1}{4}\ \frac{cm}{ft}[/tex]
That means ----> 1 cm in the drawing represent 4 ft in the mural
so
To find out the dimensions in the drawing, multiply the actual dimension by the scale drawing factor
so
[tex](16\ ft)(\frac{1}{4}\ \frac{cm}{ft})=4\ cm[/tex]
Need help doing this question! Thanks . = )
Answer:the boat will travel about 20.62 miles
Step-by-step explanation:
Since the boat travelled from dock A to dock B without passing and stopping at dock C along the way. The number of miles travelled would be the hypotenuse of the right angle triangle shown. To determine the number of miles travelled, we would apply Pythagoras theorem which is expressed as
Hypotenuse^2 = opposite side^2 + adjacent side^2
Looking at the triangle,
Opposite side = 13 miles
Adjacent side = 16 miles
Hypotenuse^2 = 13^2 + 16^2 = 425
Hypotenuse = √425 = 20.62 miles
T.J. Had a car loan for $5800. Over the course of the loan, he paid a total of $1508 in interest at a rate of 6.5%. Over how many months did T.J. Pay of the car loan?
Answer: it took T.J. 48 months to pay the car loan.
Step-by-step explanation:
The formula for simple interest is expressed as
I = PRT/100
Where
P = principal or amount loaned
R = interest rate
T = time in years.
I = interest
From the information given,
P = $5800
R = 6.5%
Over the course of the loan, he paid a total of $1508 in interest. Therefore,
I = $1508 Therefore,
1508 = (5800×6.5×T)/100
1508 = 377T
T = 1508/377 = 4 years
Assuming there are 12 months in a year, the number of months in 4 years would be 4×12 = 48 months
the x-intercepts of y=tanx are the same as the x-coordinates of the center points of y=tanx
Answer:
It is true that the x-intercepts of y=tanx are the same as the x-coordinates of the center points of y=tanx
Step-by-step explanation:
Given function is a trignometric one
y = tanx
we have tanx has values 0 for all multiples of pi.
i.e. tan x =0 whenever [tex]x = 2n\pi[/tex] for all integers n.
Also tanx has a period of pi.
It is a discontinuous graph extending in one period from -pi/2 to pi/2.
Hence the mid point of each period is the x intercept.
It is true that the x-intercepts of y=tanx are the same as the x-coordinates of the center points of y=tanx
Final answer:
The x-intercepts of the function y=tan(x) are at integer multiples of π. The concept of 'center points' for y=tan(x) is not well-defined due to the nature of the function having no central axis and being periodic.
Explanation:
The student is inquiring about the x-intercepts and center points of the function y=tan(x). The x-intercepts of the tangent function occur wherever y=0, which happens at values of x that are integer multiples of π, as the tangent function has a period of π.
On the other hand, the concept of center points is not well-defined for the tangent function since it does not have a central axis like an ellipse or a bounded pattern. The tangent function is periodic and continuous between its vertical asymptotes, which occur at odd multiples of π/2.
Clare said that (4/3) ÷ (5/2) is (10/3). She reasoned: (4/3) ⋅ 5 = (20/3) and (20/3) ÷ 2 = (10/3). Explain why Clare's answer and reasoning are incorrect. Find the correct quotient.
Answer:
Correct quotient =
[tex]\dfrac{8}{15}[/tex]
Step-by-step explanation:
We are given he following in the question:
[tex]\text{Clare's calculation:}\\\\\dfrac{4}{3}\div \dfrac{5}{2} = \dfrac{10}{3}\\\\\text{Clare's reason:}\\\\\dfrac{4}{3}\times 5 = \dfrac{20}{3}, \dfrac{20}{3} \div 2 = \dfrac{10}{3}[/tex]
Clare's, reason is incorrect.
The correct quotient can be calculated in the following manner:
[tex]\dfrac{4}{3}\div \dfrac{5}{2}\\\\= \dfrac{4}{3}\times \dfrac{2}{5} = \dfrac{8}{15}[/tex]
Clare simply multiplied the fraction with 5 and divided the product with 2 which is incorrect. We have to multiply the fraction with the reciprocal of divisor.
Answer:
The answer is 8/15
Step-by-step explanation:
4/3 divided by 5/2 is the same as 4/3 times 2/5
4/3 times 2/5= 8/15
8/15 is the answer
She is wrong because if you want to do it you have to multiply the fraction with 5 and then MULTIPLY it by 2 not divide it by 2.
Identify which value represents the sample mean and which value represents the claimed population mean.A) American households spent an average of about $52 in 2007 on Halloween merchandise such as costumes, decorations and candy. To see if this number had changed, researchers conducted a new survey in 2008 before industry numbers were reported. The survey included 1,500 households and found that average Halloween spending was $58 per household.B) The average GPA of students in 2001 at a private university was 3.37. A survey on a sample of 203 students from this university yielded an average GPA of 3.59 in Spring semester of 2012.
Answer:
A) the average value of Halloween spending of $52 in 2007 is claimed population mean and the value of Halloween spending $58 per household found out through survey done in 2008 is the sample mean
B) the average GPA 3.37 in 2001 is the claimed population mean and the average GPA through survey of 203 students is sample mean.
Step-by-step explanation:
'Claimed population mean' means the average value not taken through a survey of a sample size therefore in both options the average value is Claimed population mean
'Sample mean' is taken from a group of population therefore the value in both options taken through a survey of a sample population is Sample mean
A sample is a portion of a whole group. We use sample to predict data about the whole group(called population).
A) The sample mean is: average Halloween spending was $58 per household (survey in year 2008)
The population mean is: American households spent an average of about $52 in 2007
B) The sample mean is: average GPA of 3.59 (survey in 2012)
The population mean is: average GPA of students in 2001 at a private university was 3.37
What is sample mean and sample, and population mean and population?Sample is a portion researchers or any interested person or community takes out from a big group(called population) so as to predict properties of that big group.
We work on sample because big groups are sometimes too big that we can't cover it all in normal time. There are some other reasons too because of which we work on samples instead of population.
Sample mean is the mean obtained in the sample taken.
Population mean is hypothesized mean of population(since we don't know real mean of population, that's why hypothesized).
Thus, for given condition, we have:
A) The sample mean is: average Halloween spending was $58 per household (survey in year 2008)
The population mean is: American households spent an average of about $52 in 2007
B) The sample mean is: average GPA of 3.59 (survey in 2012)
The population mean is: average GPA of students in 2001 at a private university was 3.37
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Simplify √192
√83
√106
√643
√824
Answer:
[tex]\displaystyle 8\sqrt{3}[/tex]
Step-by-step explanation:
[tex]\displaystyle \sqrt{3 \times 64} = 8\sqrt{3}[/tex]
I am joyous to assist you anytime.
Answer:
the answer is: [tex]8\sqrt{3}[/tex]
good luck
Mrs. Bailey gives a test, and her students’ scores range from 30 to 70. She decides to curve the scores, so that they range from 65 to 95. Let "x" be an original score, and "y" be a curved score. Using the ordered pairs (30,65) and (70,95), write the equation in slope/intercept form that she should use to curve the test scores.
Answer:
f(x) = x*3/4 + 42.5
Step-by-step explanation:
The original difference between the pair is 70 - 30 = 40
The new difference between the pair is 95 - 65 = 30
Since the differences are not the same, Mrs Bailey must first perform a (slope) multiplication by a factor of 30/40 or 3/4
Then 30 * 3/4 = 22.5
Then she can shift the scores up by 65 - 22.5 = 42.5 in order to get the range from 65 to 95
Therefore, f(x) = x*3/4 + 42.5. We can test that
f(30) = 30*3/4 + 42.5 = 65
f(70) = 70*3/4 + 42.5 = 95
Which statements about triangle JKL are true? Select two options.
M is the midpoint of line segment KJ.
N is the midpoint of line segment JL.
MN = One-halfKJ
MN = 4.4m
MN = ML
Answer: 2nd and 3rd statement
Step-by-step explanation:
From the diagram, the 1st statement is wrong, the 2nd statement is correct
To know which of the rest of the statement is correct we have to find the value of MN
So we use the cosine rule to find the angle inside the triangle at point L
c2 = a2+b2-2abcosC
For our triangle
c= 12.8
b= 5.9+5.9=11.8
a= 3.7+3.7=7.4
C = ?
(12.8)2 = (7.4)2 + (11.8)2 - 2x11.8x7.4cosC
163.84 = 54.76+739.24 - 174.64cosC
163.84 = 194 - 174.64cosC
163.84-194 = -174.64cosC
-30.16 = -174.64cosC
-30.16/-174.64 =cosC
cosC = 0.1727
cos-1(0.1727) = 80.06
C = 80.06 degrees
So we use this value to find length MN with also cosine rule from the triangle NML
c2 = a2+b2-2abcosC
c = ?
a = 5.9
b = 3.7
C = 80.06
c2 = (5.9)2 + (3.7)2 - 2x5.9x3.7cosC
c2 = 34.81 + 13.69 - 43.66x0.1727
c2 = 48.5 - 7.54
c2 = 40.96
c = root(40.96)
c = 6.4
Which is half of KJ
So therefore, the third statement is correct
Answer:
b,c
Step-by-step explanation:
i did the test:)
Charles Stickley's gross weekly pay is $925. His earnings to date for the year total $21,045. What amount is deducted weekly for Social Security taxes, if the tax rate is 6.2%?
Final answer:
The amount deducted weekly for Social Security taxes for Charles Stickley is $57.35.
Explanation:
The student's question is regarding the calculation of Social Security taxes based on a given gross weekly pay and applying the current tax rate.
Charles Stickley's weekly Social Security tax deduction can be calculated by multiplying his gross weekly pay by the Social Security tax rate of 6.2%.
Here's how to calculate it:
Identify the gross weekly pay, which is $925 in this case.Calculate the Social Security tax deduction by multiplying $925 by the tax rate of 6.2% (or 0.062).Perform the calculation: 925 x 0.062 = $57.35.Therefore, the amount deducted weekly for Social Security taxes is $57.35.
PLEASE PLEASE HELP ASAP
Answer: basically just set them equal and solve
Step-by-step explanation: vertical angles are xongruent
A student wants to find point C on the directed line segment from A to B on a number line such that the segment is partitioned in a ratio of 3:4. Point A is at -6 and point B is at 2. The student's work is shown .
Answer: A No, the student should have added 3 + 4 to get the total number of sections, and used the fraction Three-sevenths instead of Three-fourths.
Step-by-step explanation:
For the following inequality, indicate whether the boundary line should be dashed or solid. x ≤ 5 dashed solid.
Answer:
Solid
Step-by-step explanation:
Rewriting for the sake of clarity:
1) For the following inequality, indicate whether the boundary line should be dashed or solid. x ≤ 5
2) Since it is a closed interval which includes the number 5 then this can also be written as:
[tex](-\infty,5][/tex]
3) Hence, we can graph it as a solid line crossing the point (5,0). For the x coordinate 5, is within the interval.
A chemist currently has a solution of 30% hydrochloric acid and a solution of 15% hydrochloric acid. He mixes the two solutions together to create 24 oz of a 25% hydrochloric acid solution. How many ounces of the 30% hydrochloric acid solution did he use in order to obtain the 25% solution
Answer:
16 ounces of the 30% hydrochloric acid solution is used in order to obtain the 25% solution.
Step-by-step explanation:
Let amount of 30% ounces be 'x' and that of 15% ounces by 'y'.
Given:
Total amount on mixing both the solution = 24 oz
∴ [tex]x+y=24\\x=24-y------------ 1[/tex]
Also, the total acid content in the resulting 25% solution is equal to the sum of the acid contents in 30% and 15% solutions.
∴ [tex]0.30x+0.15y=0.25(24)\\0.30x+0.15y=6-------2[/tex]
Now, plug in 'x' from equation (1) into equation (2). This gives,
[tex]0.30(24-y)+0.15y=6[/tex]
[tex]7.2-0.3y+0.15y=6[/tex]
[tex]-0.3y-0.15y=6-7.2[/tex]
[tex]-0.15y=-1.2[/tex]
[tex]y=\frac{1.2}{0.15}=8[/tex] ounces
Therefore, [tex]x=24-8=16[/tex] ounces
Hence, 16 ounces of the 30% hydrochloric acid solution is used in order to obtain the 25% solution
Solve for x in the equation x squared + 2 x + 1 = 17.
A) x = negative 1 plus-or-minus StartRoot 15 EndRoot
B) x = negative 1 plus-or-minus StartRoot 17 EndRoot
C) x = negative 2 plus-or-minus 2 StartRoot 5 EndRoot
D) x = negative 1 plus-or-minus StartRoot 13 EndRoot
Answer: B) x = negative 1 plus-or-minus StartRoot 17 EndRoot
Step-by-step explanation:
The given equation is x squared + 2 x + 1 = 17. It is written as
x^2 + 2x + 1 = 17
It is a quadratic equation. The general form of a quadratic equation is ax^2 + bx + c
Rearranging the given equation, it becomes
x^2 + 2x + 1 - 17 = 0
x^2 + 2x - 16 = 0
We will apply the general formula for quadratic equation. It is expressed as
x = [-b ± √(b^2 - 4ac)]/2a
From the equation,
a = 1
b = 2
c = -16
x = [-2 ± √(2^2 - 4×1×-16)]/2×1
x = [-2 ± √(4 + 64)]/2
x = (-2 ± √68)/2
x = (-2 ± 2√17)/2
x = (-2 + 2√17)/2 or x = (-2 - 2√17)/2
x = -1 + √17 or x = -1 - √17
Answer:
B) x = negative 1 plus-or-minus StartRoot 17 EndRoot i hope this was help
Step-by-step explanation:
There was a dog and a cat they both had walked 2 miles in the morning and in the afternoon the dog walked 3 miles how many miles did the cat and dog walk.
Step-by-step explanation:
In the morning cat and dog walked 2 miles.
Total distance traveled in the morning = 2 x 2 = 4 miles
In the afternoon the dog walked 3 miles
Total distance traveled in the afternoon = 3 miles
We need to find how many miles did the cat and dog walk.
Total distance traveled by cat and dog = Total distance traveled in the morning + Total distance traveled in the afternoon
Total distance traveled by cat and dog = 4 + 3
Total distance traveled by cat and dog = 7 miles
Please Help! Will give Brainliest for the correct answer. Given the function f(x) = x2 and k = -3, which of the following represents a vertical shift? (An explanation would be very helpful.)
1) f(x) + k
2) f(x + k)
3) f(kx)
4) kf(x)
Answer:
1) f(x) + k
Step-by-step explanation:
1) that would result in a vertical shift as the -3, would put parabola down 3
2) that would result in horizontal shift
3) That would change the original function
4) "kf" is not a function
Answer:
1) [tex]f(x)+k[/tex]Step-by-step explanation:
Givens
[tex]f(x)=x^{2}[/tex]
[tex]k=-3[/tex]
A vertical shift of this function is represeted by the first choice: [tex]f(x)+k[/tex].
Remember, when add or subtract units to the whole function, or the dependent variable (vertical axis), we are actually shifting the function vertically. So, the first option is adding units to the dependent variable because [tex]f(x)=y[/tex].
Therefore, the right answer is 1) [tex]f(x)+k[/tex]
Each day valerie charges her lunch account for her lunch. If the cost of lunch is $3 then by how much has her lunch account been impacted over a period of 15 days?
Answer: the amount by which her lunch account would have been impacted is $45
Step-by-step explanation:
Each day Valerie charges her lunch account for her lunch. If the cost of lunch is $3 then, it means that Valerie charges the lunch account with $3 daily. It means that in x days, she would have charged her lunch account with 3×x = $3x
If she charged her account for a period of 15 days, it means that x = 15. Therefore, the amount by which her lunch account would have been impacted is 3×15 = $45
As a prize a contest winner get to draw out one bill at a time from a box containing 10 five dollar bills 10 ten dollar bills and 10 twenty dollar bill the drawing end when 3 bill of the same denomination are drawn and of course the contest winner keeps whatever he was drawn. What is the largest sum of money that can be won under these condition?
The largest sum of money that can be won under these condition is $90.
To start finding the maximum able to be drawn, they contest winner would draw 3 [tex]\times[/tex] $20 bills = $60.
Prior to drawing the 3rd $20 bill, the contest winner could draw 2 of the $5 bills and 2 of the $10 bills.
This added to the $20 bills they had drawn would give:
(2 [tex]\times[/tex] $5) = $10
(2 [tex]\times[/tex] $10) = $20
Add all the ability to win the contest.
$10 + $20 + $60 = $90
Thus, the largest sum of money that can be won under these condition is $90.
If 375y=x2 and x and y are positive integers,
then which of the following must be an integer?
I. y/15
II. y/30
III. (y2)/25
a) I only
b) III only
c) I and II
d) I and III
e) I, II and III
Answer:
Option d - I and III.
Step-by-step explanation:
Given : If [tex]375y=x^2[/tex] and x and y are positive integers.
To find : Which of the following must be an integer?
Solution :
As we see all option there is a multiple of y.
So, we factoring the number 375
i.e. [tex]375=3\times 5\times 5\times 5[/tex]
[tex]375=15\times 5^2[/tex]
[tex]375=15\times 25[/tex]
In order for 375y to be a perfect square,
The prime factorization of y must contain at least one 3 and one 5.
or y must be a multiple of 15.
If y is a multiple of 15, then [tex]\frac{y}{15}[/tex] must be an integer.
and [tex]\frac{y^2}{25}[/tex] must be an integer.
Therefore, I and III will be correct i.e. option d.
Tom is saving for his new motorcycle. He has already saved $450 and continues to save $75 each month. How many months will it take to save at least 2,175?
Answer:
Tom required 23 months to save at least $2175.
Step-by-step explanation:
Given:
Money already Saved = $450
Money Needs to be saved = $2,175
Each month saving = $75
We need to find the Number of months required to save at least $2,175
Solution:
Let the number of months be 'x'.
Now We can say Total Money which is already save plus Each month saving multiplied by number of months should be equal to Money Needs to be saved.
Framing in equation form we get;
[tex]450+75x=2175[/tex]
Solving the equation we get;
We will Subtract both side by 450 we get;
[tex]450+75x-450=2175-450\\\\75x=1725[/tex]
Now By dividing 75 on both side we get;
[tex]\frac{75x}{75}= \frac{1725}{75}\\\\x= 23\ months[/tex]
Hence Tom required 23 months to save at least $2175.
The sum of the first three terms of a geometric progression is 126. If 14, 36, and 4 are added to each of these terms, respectively, then the new numbers form an arithmetic progression. Find the sixth term of the geometric progression
Answer:
if r=4:
[tex]\displaystyle a_6=6144[/tex]
if r=1/4:
[tex]\displaystyle a_6=\frac{3}{32}[/tex]
Step-by-step explanation:
Geometric and Arithmetic Progressions
We define a geometric progression when each term [tex]a_n[/tex] is defined as the previous term [tex]a_{n-1}[/tex] times a constant called the common ratio. The iterative formula is
[tex]\displaystyle a_n=a_1.r^{n-1}[/tex]
In an arithmetic progression, each term is found by adding a constant called common difference, to the previous term
[tex]\displaystyle a_n=a_1+(n-1).r[/tex]
We are given the condition that the sum of the three first terms of a geometric progression is 126
[tex]\displaystyle a_1+a_2+a_3=126[/tex]
Using the iterative formula, we have
[tex]\displaystyle a_1+a_1.r+a_1.r^2=126[/tex]
Taking a common factor
[tex]\displaystyle a_1(1+r+r^2)=126....[eq\ 1][/tex]
We also know that if 14, 36, and 4 are added to each term, respectively, the new numbers form an arithmetic progression. It means they will have a common difference. The new numbers will be
[tex]\displaystyle a_1'=a_1+14[/tex]
[tex]\displaystyle a_2'=a_2+36[/tex]
[tex]\displaystyle a_3'=a_3+4[/tex]
The common difference between term 2 and term 1 is
[tex]\displaystyle a_2'-a_1'=a_2+36-a_1-14[/tex]
Using the iterative formula again
[tex]\displaystyle a_2'-a_1'=a_1.r-a_1+22[/tex]
The common difference between term 3 and term 2 is
[tex]\displaystyle a_3'-a_2'=a_3+4-a_2-36[/tex]
Using the iterative formula again
[tex]\displaystyle a_3'-a_1'=a.r^2-a.r-32[/tex]
Both common differences must be equal
[tex]\displaystyle a_1.r-a_1+22=a_1.r^2-a_1.r-32[/tex]
Rearranging
[tex]\displaystyle 2a_1r-a_1r^2-a_1=-54[/tex]
Solving for [tex]a_1[/tex]
[tex]\displaystyle a_1=\frac{54}{1-2r+r^2}......[eq\ 2][/tex]
Replacing in eq 1
[tex]\displaystyle \frac{54(1+r+r^2)}{1-2r+r^2}=127[/tex]
Dividing by 18 and cross-multiplying
[tex]\displaystyle 3+3r+3r^2=7-14r+7r^2[/tex]
Rearranging we have a second-degree equation
[tex]\displaystyle 4r^2-17r+4=0[/tex]
Factoring
[tex]\displaystyle (r-4)(4r-1)=0[/tex]
The solutions are
[tex]\displaystyle r=4\ ,\ r=\frac{1}{4}[/tex]
If r=4, and using eq 2
[tex]\displaystyle a_1=\frac{54}{1-8+16}=6[/tex]
Having [tex]a_1[/tex] and r, we compute [tex]a_6[/tex]
[tex]\displaystyle a_6=a_1.r^5=6.(4)^5[/tex]
[tex]\displaystyle a_6=6144[/tex]
If we use the other solution r=1/4
[tex]\displaystyle a_1=\frac{54}{1-\frac{1}{2}+\frac{1}{16}}=\frac{54}{\frac{9}{16}}[/tex]
[tex]\displaystyle a_1=96[/tex]
The sixth term is
[tex]\displaystyle a_6=96(\frac{1}{4})^5=\frac{96}{1024}[/tex]
[tex]\displaystyle a_6=\frac{3}{32}[/tex]
Both solutions are feasible
Final answer:
In this question, we explore geometric and arithmetic progressions to find the 6th term of a geometric progression given specific conditions.
Explanation:
Geometric Progression (GP): In a GP, the sum of the first three terms is 126. Let the first term be a and the common ratio be r. The sum of the first three terms can be represented as a + ar + ar^2 = 126.
Arithmetic Progression (AP): Adding 14, 36, and 4 to the terms of the GP forms an AP. The common difference of this AP is 36 - 14 = 22.
Finding the 6th Term: Using the formula for the nth term of a GP, the 6th term can be calculated as a * r^5.
Determine whether the function below is an even function, an odd function, both, or neither.
f(x)=(x+5)^2
A.
neither even nor odd
B.
even function
C.
both even and odd
D.
odd function
Answer:
A. neither even nor odd
Step-by-step explanation:
The equation is that of a parabola whose line of symmetry is x=-5. Even functions are symmetrical about the line x=0, so this is not an even function. It has terms of even degree, so is not an odd function.
The function is neither even nor odd.
Answer:
Option A - neither even nor odd
Step-by-step explanation:
Given : [tex]f(x)=(x+5)^2[/tex]
To find : Determine whether the function below is an even function, an odd function, both, or neither ?
Solution :
We know that,
1) If f(-x)=f(x) it is an even function.
2) If f(-x)=-f(x) it is a odd function.
[tex]f(x)=(x+5)^2[/tex]
[tex]f(x)=x^2+10x+25[/tex]
Substitute x with -x in the function,
[tex]f(-x)=(-x+5)^2[/tex]
[tex]f(-x)=x^2-10x+25[/tex]
The function does not comply with the definitions.
The function is neither even nor odd.
Therefore, option A is correct.