In a 30-60-90 triangle:
[tex]\begin{gathered} Hypotenuse_{\text{ }}=_{\text{ }}2\cdot short_{\text{ }}length \\ Long_{\text{ }}length=\sqrt[]{3}\cdot short_{\text{ }}length \\ \end{gathered}[/tex]so:
[tex]\begin{gathered} Long_{\text{ }}length=\sqrt[]{3}\cdot5 \\ Long_{\text{ }}length=5\cdot\sqrt[]{3}\approx8.66 \end{gathered}[/tex]The table below gives the population of a town (in thousands) from 2000 to 2008. What was the average rate of change of population between 2002 and 2004, and between 2002 and 2006?
The average rate of change in population is -35 between 2002 and 2004, and -2.5 between 2002 and 2006 which represents the decrease in the population.
What is Lagrange mean value theorem?Lagrange mean value theorem states that, if a function f is continuous over the closed interval [a, b] and differentiable over the open interval (a, b), then there must be at least one point c in the interval (a, b) where the slope of the tangent at the point c is equal to the slope of the tangent through the curve's endpoints, resulting in the expression f'(c) = {F(b) -F(a)}/(b-a)
As per the given data in the table, the required solution would be below
The average rate of change in population between 2002 and 2004 as:
⇒ (76-83)/(2004-2002)
⇒ -7/2
⇒ -3.5
The average rate of change in population between 2002 and 2006
⇒ (78-83)/(2006-2002)
⇒ -5/2
⇒ -2.5
This represents the decrease in the population.
Therefore, the average rate of change in population is -35 between 2002 and 2004, and -2.5 between 2002 and 2006.
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What's the value of b ? See attached screenshot.
The value of b would be 25/4.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Since the equation of the line is 2y = 4.5, where c is a constant, the y-coordinate of the intersection point must be c.
The parabola has a equation y = -4x² + bx, where bis a positive constant.
The solution to this quadratic equation will gives the x-coordinate(s) of the point(s) of intersection
Since it’s given that the line and parabola intersect at exactly one point, the equation y = -4x² + bx has exactly one solution.
A quadratic equation in the form ax²+bx+c has exactly one solution when its discriminant b²−4ac is equal to 0.
Therefore, if the line y = 22.5 intersects the parabola defined by exactly one point, then by = 25/4 .
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Which of the following functions grows the fastest as x grows without bound?
Answer:
[tex]f(x)=e^x[/tex]Explanation:
First, the function, g(x):
[tex]g(x)=e^{\cos(x)}[/tex]The function g(x) oscillates, thus, it does not increase.
The value of 'e' is approximately 2.7.
[tex]\begin{gathered} f(x)=e^x\approx2.7^x \\ h(x)=(2.5)^x \end{gathered}[/tex]Since 2.7 is greater than 2.5, we can infer that f(x) grows the fastest as x grows without bound.
find the exact value of cos90°.
cos90° is equal zero
A manufacturing process produces a critical part of average length 90 millimeters, with a standard deviation 2 of millimeters. All parts deviating by more than 5 millimeters from the mean must be rejected. What percentage of the parts must be rejected, on average? Assume a normal distribution.
We have that
[tex]X\sim N(\mu=90,\sigma^2=4^{})[/tex]The parts the will be rejected when it's above 95 or when it's under 85, if we plot the normal distribution it would be
Then, the percentage of the parts that will be rejected corresponds to the area in blue, then, we must calculate the area under the normal distribution for
[tex]P(X<85)+P(X>95)[/tex]The normal distribution is symmetrical, then calculate P(X < 85) is the same as P(X > 95), then we write it as
[tex]2\cdot P(X>95)[/tex]Calculate that integral is very hard, then, we must transform that in a standard normal X ~ N(0, 1) and use a table to find the result, to do that we must write a value z, it's a transformation to take a value on our normal and leads it to the standard normal, it's
[tex]Z=\frac{X-\mu}{\sigma}[/tex]We have X = 95, μ = 90 and σ = 2
[tex]Z=\frac{95-90}{2}=2.5[/tex]Then 2.5 is the value we are going to search in our table, using the complementary cumulative table for 2.5 we get 0.00621, which means
[tex]P(X>95)=0.00621[/tex]And the total percentage will be
[tex]P(X<85)+P(X>95)=0.01242[/tex]We can write it in percentage
[tex]0.01242=1.242\%[/tex]Therefore, only 1.24% will be rejected.
It's a very low value, but it's expected because it's more than 2 standard deviations (95%).
5. GMAT scores are approximately normally distributed with a mean of 547 and a standard deviation of 95. Estimate the percentage of scores that were(a) between 357 and 737. %(b) above 737. %(c) below 452. %(d) between 452 and 737. %
Problem Statement
The question tells us that the GMAT scores are approximately normally distributed with a mean of 547 and a standard deviation of 95.
We are asked to find the percentage of scores that were:
a) between 357 and 737.
b) above 737
c) below 452
d) between 452 and 737.
Solution
a) Between 357 and 737:
[tex]\begin{gathered} 357\text{ is 2 standard deviations less than the mean of 547. That is,} \\ 547-2(95)=357 \\ \text{This means that 357 is }\frac{95}{2}\text{ \% from the mean}=47.5\text{ \% from 547.} \\ \\ 737\text{ is 2 standard deviations greater than the mean of 547. That is,} \\ 737-2(95)=547. \\ \text{This means that 737 is }\frac{95}{2}\text{ \% from the mean }=47.5\text{ \% from 547} \\ \\ \text{Thus the range 'Between 357 and 737' is:} \\ (47.5+47.5)\text{ \%}=95\text{ \%} \end{gathered}[/tex]b) Above 737
[tex]\begin{gathered} 737\text{ is 2 standard deviations away from the mean as shown in question A.} \\ \text{Thus, the percentage of scores above 737 must be:} \\ 100\text{ \% - (50 + 47.5)\% }=2.5\text{ \%} \end{gathered}[/tex]c) Below 452:
[tex]\begin{gathered} 452\text{ is 1 standard deviation from the mean.} \\ \text{Thus the percentage of scores below 452 must be:} \\ 50\text{ \% - 34\% = 16\%} \end{gathered}[/tex]d) Between 452 and 737:
[tex]\begin{gathered} 452\text{ is 1 standard deviation lower than the mean 547. Thus, the percentage from 452 to 547 is 34\%} \\ 737\text{ is 2 standard deviations higher than the mean of 547. Thus the percentage from 547 to 737 is: 47.5\%} \\ \\ \text{Thus the percentage between 452 and 737 is: (34 + 47.5)\%= 81.5\%} \end{gathered}[/tex]Hidden Hollow Mining Co. acquired mineral rights for $49,500,000. The mineral deposit is estimated at 55,000,000 tons. During the current year, 17,050,000 tons were mined and sold.
A. Determine the depletion rate.B. Determine the amount of depletion expense for the current year.C. Journalize the adjusting entry on December 31 to recognize the depletion expense. Refer to the Chart of Accounts for exact wording of account titles.
A. The depletion rate is $0.90 per ton.
B. The amount of depletion expense that Hidden Hollow Mining Co. should recognize for the current year is $15,345,000.
C. The adjusting journal entry to recognize the depletion expense is as follows:
Adjusting Journal Entry:Debit Depletion Expenses $15,345,000
Credit Accumulated Depletion $15,345,000
To recognize the depletion expense for the current year.What is an adjusting entry?Adjusting entries are the journal entries made at the end of the accounting period to recognize unrecorded expenses, revenue, and other gains or losses.
An example of an adjusting entry is the entry to recognize depreciation, amortization, or depletion expenses.
The value of the mineral rights = $49,500,000
The estimated mineral deposit = 55,000,000 tons
Depletion rate = $0.90 per ton ($49,500,000/55,000,000)
The current year's tons mined and sold = 17,050,000 tons
Amount of depletion expense for the current year = $15,345,000 (17,050,000 x $0.90)
Adjusting Entry Analysis:Depletion Expenses $15,345,000 Accumulated Depletion $15,345,000
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Aurora raised money for a white water rafting trip Jacy made the first donation Gillermo’s donation was twice Jacy’s donation Rosa’s mother tripled what Aurora had raised so far now Aurora has $120. how much did Jacy donate?
Help please!!!
The amount of money donated by Jacy is $10.
Given that, Jacy made the first donation Gillermo’s donation was twice Jacy’s donation Rosa’s mother tripled what Aurora had raised so far now Aurora has $120.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Let the donation given by Jacy be $x.
Divide that amount by 4. One part is Guillermo's and Jacy's donation and three parts is the amount donated by Rosa's mother = 120 ÷ 4 = $30
Divide that amount by 3. One part is for Jacy's donation ($x) and two part is the amount Guillermo donated x = 30/3 = $10
Jacy was the first to donate. So, Jacy donated is $10
Therefore, the amount of money donated by Jacy is $10.
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I need help with this question
The coordinates of B after the translation is ( 3 , -2)
What is coordinates in maths?
Coordinates are a pair of integers (also known as Cartesian coordinates), or occasionally a letter and a number, that identify a specific point on a grid, also known as a coordinate plane. The x axis (horizontal) and y axis are the two axes that make up a coordinate plane (vertical).Transformation involves changing the position of a shape.
The coordinates of B after the translation is (-4,-2)
A = ( 1, 1)
B = ( 3 , 4)
C = ( -1 , 8 )
The translation rule is given as ( x , y-6)
So, the coordinates of B' is calculated using
( x , y ) ⇒ ( x , y-6)
This gives
B = ( 3 , 4) ⇒ ( 3 , 4 - 6 )
( 3 , 4) ⇒ ( 3, -2 )
Rewrite as
B' = ( 3, -2 )
Hence, the coordinates of B after the translation is ( 3 , -2)
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I need help with this and please get this one right
There is a 0.765 percent chance that the flight will leave on time when it is not raining.
What is probability?Gonna determine how likely something is to occur, use probability. Many things are hard to predict with 100% certainty. We can only anticipate the possibility of an event occurring using it, or how likely it is. In the probability scale, 0 indicates an impossibility and 1 indicates a certainty.
Given a 0.1 delay probability, the probability of the airplane departing on time is 1-0.1 = 0.9.
The likelihood that it won't rain is 1-0.15 = 0.85.
If it weren't raining, there is a 0.9 (0.85) = 0.765 percent chance that a flight would leave on time.
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A box has length 4 ft, width 5 ft, and height 6 ft. What is the volume?
The volume of box will be 120 ft.³
What is volume ?
Volume is a three dimensional space occupy by the body of particular shape such as here :
Volume of cuboidal box = lbh
where, length "l" = 4 ft.
width "b" = 5 ft.
height "h" = 6 ft.
now, the volume of box will be :
V = lbh
V = 4 x 5 x 6
V = 120 ft.³
Therefore, the volume of box will be 120 ft.³
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In earn contains two white marbles, three green marbles, and 5 red marbles. A marble is drawn and then replaced. Then the second marble is drawn. What is the probability that the first marbel was white and the second was Green?
total number of outcomes = 2 + 3 + 5 = 10
The probability of getting a white marble is:
[tex]P(white)=\frac{2}{10}=\frac{1}{5}[/tex]The probability of getting a green marble is:
[tex]P(green)=\frac{3}{10}[/tex]The events: getting a white marble and getting a green marble are independent since there is a replacement after each drawing. Then, the probability that the first marble was white and the second was Green is:
[tex]\text{ P(white and gr}een\text{) =}P(white)\cdot P(green)=\frac{1}{5}\cdot\frac{3}{10}=\frac{3}{50}[/tex]A hammock is suspended between two trees. The curve the hammock makes can bemodelled by the equation y = 0.2x² - 0.4x - 0.6, where x and y are measured inmetres.a) Find the x interceptsb) Find the vertex.c) What is the minimum height of the hammock?
We have the function that relates x and y expressed as:
[tex]y=0.2x^2-0.4x-0.6[/tex]a) We have to find the x-intercepts.
To do that we can use the quadratic equation:
[tex]\begin{gathered} x=\frac{-(-0.4)\pm\sqrt{(-0.4)^2-4(0.2)(-0.6)}}{2(0.2)} \\ x=\frac{0.4\pm\sqrt{0.16+0.48}}{0.4} \\ x=\frac{0.4\pm\sqrt{0.64}}{0.4} \\ x=\frac{0.4\pm0.8}{0.4} \\ x=1\pm2 \\ x_1=1-2=-1 \\ x_2=1+2=3 \end{gathered}[/tex]Then, we have x-intercepts at x = -1 and x = 3.
b) We have to find the vertex.
We can find the x-coordinate of the vertex using the linear coefficient b = -0.4 and the quadratic coefficient a = 0.2:
[tex]x_v=\frac{-b}{2a}=\frac{-(-0.4)}{2(0.2)}=\frac{0.4}{0.4}=1[/tex]It can also be calculated as the average of the x-intercepts.
Knowing the x-coordinate of the vertex, we can find the y-coordinate of teh vertex using the formula applied to x = 1:
[tex]y=0.2(1)^2-0.4(1)-0.6=0.2-0.4-0.6=-0.8[/tex]Then, the vertex is (1, -0.8).
c) The minimum height will be given by the y-coordinate of the vertex.
Relative to the horizontal axis (y = 0), the minimum height will be -0.8 meters below that level.
Answer:
a) The x-intercepts are x = -1 and x = 3.
b) The vertex is (1,-0.8)
c) The minimum height is 0.8 units below the horizontal axis.
Use a number line to round the number 573 to the nearest 10.
Answer: The answer is 570.
Step-by-step explanation:
So, if you know what numbers are, when you round, this question is easy. If you take a number line and put 570 on the left side and 580 on the right side, and 575 in the middle, you almost got it. Since the number we want to round is 573, we put it in the middle of 570 and 575. And since the number is closer to 570 than 580, the answer is 570. Your welcome!
What equation can be written in a form that shows a proportional relationship using variables
Answer:
y = kx
Step-by-Step explanation:
An equation shows a proportion relationship when it can be written in the following format:
[tex]\frac{y}{x}[/tex]So, in the context of this exercise, we need to pass x to the other side of the equation dividing, by itself.
In this question:
y = k+x
x alone cannot be passed dividing, just k+x
Y = k, there is no x.
The answer is y = kx, because we can also write this as:
[tex]\frac{y}{x}=k[/tex]9-3 2 = (x) 6 what is the function family.
a. G(x) = 1/4x - 5 Since it's similar to the equation f(x)=mx + b, which is a linear
function, g(x) is a linear function.
b. f(x) = 2*(x - 1)^2 - 5 Since the degree of the polynomial is 2, we deduce it is a cuadratic function.
c. f(x) = 7 Since the degree of x is zero ( there is no x) , we deduce it is a constant function.
What does it mean to take a derivative? I know how it's done, but not why.
Finding the derivative of the function is just a way for us to discuss how the function changes. For example, if we want to get the derivative of function y, with respect to x (dy/dx), then it is a formal way of discussing, how y changes when x changes.
The table shows the amount of water Joel had in his bathtub to wash his dog and his desired water level. If the water drains at a rate of 14 gallons per minute, how many minutes will it take the tub to drain to his desired level?
Starting Water Level = 42 gallons
Desired Water Level = 28 gallons
It will take 1 minute to tub to drain to his desired level, by Rate of change.
What is rate of change?
Rate of change is used to mathematically describe the percentage change in value over a defined period of time.
Given, starting water value = 42 and desired water level = 28
Rate of change = 14 gallons.
Let x be the time,
According to question,
42-14x=28
-14x=-14
x=1
Hence, it will take one minute.
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Use the properties of exponents to write an equivalent expression for each given expression.
1. 6^4 x 6^3
2. (3^6)^-2
3. 7^3 x 2^3
4. 4^10 divided by 4^4
Please help :)
Answer:
Step-by-step explanation:
is the answer 9 im lost can you help me
Solution
Population of the town = 3000
Rate = 4%
Amount = 4700
[tex]A=P(1+r)^n[/tex][tex]\begin{gathered} 4700=3000(1+4\text{ \%\rparen}^n \\ \frac{4700}{3000}=1+0.04)^n \\ 1.567=1.04^n \end{gathered}[/tex]Find log of both side
[tex]\begin{gathered} 1.567=1.04^n \\ ln1.567=ln1.04^n \\ nln1.04=1.567 \\ n=\frac{ln1.567}{ln1.04} \\ n=11.5yrs \end{gathered}[/tex]Therefore the number of years it will take population to reach 4700 = 11.5yrs
Mr. Knox's garden is in the shape of a triangle. What is the area of Mr. Knox’s garden? __ square feet
Using the area of a triangle, the area of Mr. Knox's garden is: 58 square feet.
How to Find the Area of a Triangular Shape?The area of any triangular shape can be calculated using the formula for the area of a triangle.
The area of a triangle = 1/2(base)(height).
The shape of Mr. Knox's garden is triangular. The parameters of his triangular garden are:
Length of the base of the triangular garden = 24½ ft
Height of the triangular garden = 8 ft.
Area of the garden = 1/2(24½)(8)
Area of the garden = 1/2(29/2)(8)
Area of the garden = 1/2(232/2)
Area of the garden = 232/4
Area of the garden = 58 square feet.
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The length of a bridge is about 6.3 meters. Which number from the list is closest to 6.3?A. 40B. 748C. 735D. 737Pls help me
The closest number to 6.3 is the number 40, because when we substract 6.3 from 40, we get the smallest number.
40 - 6.3 = 33.7
748 - 6.3 = 741.7
Consider the following compound inequality. 7
a)The solution of the inequality is 3 < x ≤ 5
c) The solution in interval notation is (3, 5]
Step - by - Step Explanation:
What to find?
• The solution of the inequality.
,• The graph of the inequality.
,• The solution in interval notation.
Given:
7 < x +4 ≤ 9
Re-write the above inequality.
x + 4 > 7 or x + 4 ≤ 9
Solve for x in each case.
x + 4 > 7
Subtract 4 from both-side of the inequality.
x > 7 - 4
x > 3
x + 4 ≤ 9
Subtract 4 from both-side of the inequality.
x ≤ 9 - 4
x≤ 5
Combine the two solutions.
3Hence, the solution to the inequality is 3 < x ≤ 5
b) We can proceed to graph the inequality.
c) The solution in interval notation is (3, 5]
A cube-shaped box has side lengths of 1.5 m, and it exerts a force of 63 N on the ground. Calculate the pressure, in N/m², that the box exerts on the ground. If your answer is a decimal, give it to 1 d.p.
Answer:
28 N/m²
Step-by-step explanation:
Since you are calculating pressure, you want your question in N/m².
Note that the one side of the box has a length of 1.5m, and so to get the area of the bottom face, you need to square 1.5m::
(1.5m)² = 2.25m²
Then you need to use the formula P=F/A
Where P is pressure, F is force, and A is area, plug in your variables::
P = [tex]\frac{63N}{2.25m^{2} }[/tex]
Then you get an answer of:[tex]28 \frac{N}{m^{2} }[/tex]
You always want to make sure that the answer's units align with what you are told to solve for. In this case, they do, so no further steps are needed.
Hope this helps! :)
why is 5.1 bigger than 5.099
5.1 is greater than 5.099, because the value of the 1 in 5.1 is more than the value of the 99 in 5.099.
[tex]\begin{gathered} 5.1=5+0.1=5+\frac{1}{10}=5+\frac{100}{1000} \\ 5.099=5+0.099=5+\frac{99}{1000} \end{gathered}[/tex]The value of 1 in 5.1 is 0.1, while the value of the 99 in 5.099 is 0.099.
Since 0.1 is bigger than 0.099, then 5.1 is bigger than 5.099.
Also, 5.1 is greater than 5.099 because a bigger number on 5.1 (which is 1) is closer to the decimal point compared to 5.099 (1 is bigger than 0). the closer a decimal is to the decimal point the higher its value.
Fay is paid semimonthly. The net amount of each paycheck is $670.50.What is her net annual income?a. $17,433b. $4,023c. $16,092d. $8,046
Answer:
c. $16,092
Explanation:
• Fay is paid semimonthly, that is, ,twice a month,.
,• There are ,12 months in a year,.
Thus, the number of paychecks she receives annually is: 2 x 12 = 24.
The net amount of each paycheck is $670.50.
In order to get her net annual income, multiply the net amount on each paycheck by the number of payments.
[tex]\text{Net Annual Income}=24\times670.50=\$16,092[/tex]Fay's net annual income is $16,092.
Option C is correct.
Passing through (-4,-5) and parallel to the line whose equation is y= -3x+4
For the given values the equation of new line is y = -3x - 17
What is a line equation example?The formula for these lines is y = mx + b, where m denotes the slope and b the y-intercept. Our line has a slope of 3 and a y-intercept of -5, which we know from the question. By entering these numbers, we obtain the equation of our line as y = 3x - 5.
Given,
The equation of line is y= -3x+4
as the lines are parallel,
slope of lines will be same,
∴ Slope of new line = -3
The equation of line is:
y - y1 = m(x - x1)
Where x1, y1 are the given coordinates and m is the slope
⇒ y -(-5) = -3 (x - (-4))
⇒ y + 5 = -3 (x + 4)
⇒ y + 5 = -3x - 12
⇒ y = -3x - 17
∴ The equation of the line is y = -3x - 17
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4.
The Freshman Class treasury has 30
ten- and twenty-dollar bills that have
a total value of $430. How many of
each bill do they have?
There are 13 $20 bills and 17 $10 bills, respectively.
A linear equation is what?Constants and variables are used in conjunction to create linear equations. A linear equation with one variable is shown in the following standard form: Where a 0 and x is the variable, ax + b = 0.
Due to that,
There are 30 bills in all.
Total = $430
Let,
x = the number of $20 bills.
Amount in $10 banknotes = (30-x)
20x+10(30-x) = 430
20x+300-10x = 430
10x = 430-300
10x = 130
x = 13
$20 bills: x = 20; y = 13.
30 x = 30 13 = 17 = number of $10 banknotes
Therefore, there are 13 $20 bills and 17 $10 banknotes.
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A rancher has 800 feet of fencing to put around a rectangular field and then subdivide the field into three identical smaller rectangular plot by placing to fence is parallel to the field shorter side. Find the dimensions that maximize the enclosed area. Write your answer as a fraction reduced to lowest term
The diagram of the problem is:
S is the length of the shorter side of the fence. L is the length of the longest side of the field.
We know that the perimeter of the rectangle is 800ft. This means:
[tex]2S+2L=800[/tex]And the area:
[tex]A=SL[/tex]The smaller rectangles will have dimensions:
The area is:
[tex]a=\frac{SL}{3}[/tex]As we can see, if we maximize the area of the bigger rectangle "A", we are also maximizing the area of the smaller rectangles "a".
Then, we have two equations:
[tex]\begin{gathered} 2S+2L=800 \\ A=SL \end{gathered}[/tex]We can solve for L in the first equation:
[tex]\begin{gathered} 2S+2L=800 \\ 2L=800-2S \\ L=400-S \end{gathered}[/tex]Then substitute in the second:
[tex]A=S(400-S)[/tex]Simplify:
[tex]A=400S-S^2[/tex]This is a function of the area depending on the length of the shorter side of the rectangle:
[tex]A(S)=400S-S^2[/tex]We can find the maximum of this function if we find the value where the derivative of this function is 0.
Let's differentiate:
[tex]A^{\prime}(S)=400-2S[/tex]And now we find where A'(S) = 0:
[tex]\begin{gathered} 0=400-2S \\ 2S=400 \\ S=200 \end{gathered}[/tex]We have found that the shorter side must have a length of 200ft to maximize the area. Let's find the length of the larger side:
[tex]L=400-200=200[/tex]As expected, the quadrilateral which maximizes the area is the square. Thus, the dimensions of the field are 200ft x 200ft
see attached graph photo
Given: The square STUV
To Determine: The coordinate of the image after reflection over the y-axis
Solution
The reflection over the y-axis rule is
[tex](x,y)\rightarrow(-x,y)[/tex]Locate the coordinates of STUV
Let us apply the rule to get the coordinate of the image
[tex]\begin{gathered} S(-10,-10)\rightarrow S^{\prime}(10,-10) \\ T(0,-10)\rightarrow T^{\prime}(0,-10) \\ U(0,0)\rightarrow U^{\prime}(0,0) \\ V(-10,0)\rightarrow V^{\prime}(10,0) \end{gathered}[/tex]Hence, the coordinate of the image after a reflection over the y-axis is
S'(10, - 10)
T'(0, - 10)
U'(0, 0)
V'(10, 0)