Rectangle 1 has length x and width y. Rectangle 2 is made by multiplying each dimension of Rectangle 1 by a
factor of k, where k>0.
(a) Are Rectangle 1 and Rectangle 2 similar? Why or why not?
(b) Write a paragraph proof to show that the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1.
(c) Write a paragraph proof to show that the area of Rectangle 2 is k2 times the area of Rectangle 1.
Answers
Answer:
(a) Rectangle 1 and rectangle 2 are similar
(b) The perimeter of Rectangle 2 is k times the perimeter of Rectangle 1
(c) The area of Rectangle 2 is k² times the area of Rectangle 1
Step-by-step explanation:
* Lets talk about the similarity
- Two rectangles are similar if there is a constant ratio between
their corresponding sides
- Rectangle 1 has dimensions x and y
- Rectangle 2 has dimensions kx and ky
- The ratio between their dimensions is:
kx/x = k and ky/y = k, so there is a constant ratio K between their
corresponding dimensions
(a) Rectangle 1 and rectangle 2 are similar
- The perimeter of any rectangle is 2(the sum of its two dimensions)
∵ Rectangle 1 has dimensions x and y
∴ Its perimeter = 2(x + y) = 2x + 2y ⇒ (1)
∵ Rectangle 2 has dimensions kx and ky
∴ Its perimeter = 2(kx + ky) = 2kx + 2ky
- By taking k as a common factor
∴ Its perimeter = k(2x + 2y) ⇒ (2)
- From (1) and (2)
∵ The perimeter of rectangle 1 = (2x + 2y)
∵ The perimeter of rectangle 2 = k(2x + 2y)
∴ The perimeter of rectangle 2 is k times the perimeter of rectangle 1
(b) The perimeter of Rectangle 2 is k times the perimeter of Rectangle 1
- The area of any rectangle is the product of its two dimensions
∵ Rectangle 1 has dimensions x and y
∴ Its area = x × y = xy ⇒ (1)
∵ Rectangle 2 has dimensions kx and ky
∴ Its area = kx × ky = k²xy ⇒ (2)
- From (1) and (2)
∵ The area of rectangle 1 = xy
∵ The area of rectangle 2 = k²xy
∴ The area of rectangle 2 is k² times the area of rectangle 1
(c) The area of Rectangle 2 is k² times the area of Rectangle 1
(a) Yes, Rectangle 1 and Rectangle 2 are similar. (b) It is proved that the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1. (c) It is proved that the area of Rectangle 2 is [tex]k^2[/tex] times the area of Rectangle 1.
(a) Yes, Rectangle 1 and Rectangle 2 are similar. They are similar because each dimension of Rectangle 2 is k times the corresponding dimension of Rectangle 1, which means that all angles remain the same and the sides are in proportion.
(b) To prove that the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1, we start by noting that the perimeter of a rectangle is the sum of all its sides. For Rectangle 1, the perimeter [tex]P_1[/tex] is given by :
[tex]P_1 = 2(x + y)[/tex]
For Rectangle 2, each dimension is multiplied by k, so the length becomes kx and the width becomes ky. Therefore, the perimeter [tex]P_2[/tex] of Rectangle 2 is:
[tex]P_2 = 2( kx + ky)[/tex]
[tex]P_2 = 2k(x + y)[/tex]
Since 2(x + y) is the perimeter of Rectangle 1, we can replace it with [tex]P_1[/tex], yielding [tex]P_2 =[/tex] [tex]kP1[/tex] . This shows that the perimeter of Rectangle 2 is indeed k times the perimeter of Rectangle 1.
(c) To prove that the area of Rectangle 2 is [tex]k^2[/tex] times the area of Rectangle 1, we consider the formula for the area of a rectangle, which is the product of its length and width.
The area [tex]A_1[/tex] of Rectangle 1 is:
[tex]A_1 = xy[/tex]
For Rectangle 2, the length and width are both multiplied by k, so the area [tex]A_2[/tex] is:
[tex]A_2 = (kx)(ky)[/tex]
When we multiply these out,
we get:
[tex]A2 =[/tex] [tex]k^2xy[/tex]
Since xy is the area of Rectangle 1, we can replace it with [tex]A_1[/tex], giving us [tex]A_2 =[/tex] [tex]k^2A_1[/tex] . This demonstrates that the area of Rectangle 2 is [tex]k^2[/tex] times the area of Rectangle 1.
The diagram of rectangles:
Related Questions
The odometer of a new automobile functions improperly and registers only 2 miles for every 3 miles driven. If the odometer indicates 48 miles, how many miles has the automobile actually been driven?
Answers
Final answer:
The odometer reads 2 miles for every 3 actual miles driven. To find the actual distance when the odometer shows 48 miles, a proportion is used, leading to the calculation that the automobile has driven 72 actual miles.
Explanation:
The student is dealing with a proportion problem in mathematics where the automobile's odometer is inaccurate. Given that the odometer registers 2 miles for every 3 miles driven, we can set up a proportion to find the actual distance driven. If the odometer indicates 48 miles, we can write the proportion as follows:
2 miles (odometer) / 3 miles (actual) = 48 miles (odometer) / x miles (actual)
Cross-multiply to solve for x: 2x = 48 × 3
Calculate x: 2x = 144
Divide both sides by 2 to isolate x: x = 144 / 2
Find the value of x: x = 72 miles
Therefore, the automobile has actually been driven 72 miles.
When it goes on sale for $1,600, Mikah buys a new plasma television that is valued at $1,800. He takes $300 from his savings account to use as a down payment and finances the rest
Answers
Mikah's assets increased by $1,500.
Answer: B
Step-by-step explanation:
2. Tell which measure of central tendency best describes the data
3. Tell which measure of central tendency best describes the data
Answers
Answer:
Step-by-step explanation:
Mean, it is the average of all numbers and most accurately represents the data set.
The best measure of central tendency depends on the data set characteristics: the mean is suitable for outlier-free data, the median is better for skewed data or when outliers are present, and the mode indicates the most common value.
Explanation:Choosing the Best Measure of Central Tendency
When analyzing a data set, selecting the appropriate measure of central tendency is crucial. The three main measures are the mean, median, and mode. The mean is the arithmetic average and is best used when the data set does not contain outliers that can skew the results. The median is the middle value when the data is ordered and offers a better representation when outliers are present. The mode is the most frequently occurring value and provides insight into the most common data point.
In a symmetrical data set where the data are evenly distributed, all three measures of central tendency—the mean, median, and mode—will be equal or close to each other. In a skewed distribution or when the data contains outliers, the median is generally the most reliable measure of the center, as it is unaffected by extreme values.
Sandy mixed together 9 gallons of one brand of juice and 8 gallons of a second brand of juice which contains 48% real fruit juice. Find the percent of real fruit juice in the first brand if the mixture contained 30% real fruit juice.
A) 18%
B) 140%
C) 180%
D) 14%
Answers
Answer:
D) 14%
Step-by-step explanation:
Let the percent of the real fruit in the first brand = x
Given that number of gallons of real fruit in the first brand = 9 gallons
Given that percent of the real fruit in the second brand = 48%
Given that number of gallons of real fruit in the second brand = 8 gallons
Given that percent of the real fruit in the mixture = 30%
then number of gallons of real fruit in the mixture = (9+8) = 17 gallons
Then we get equation:
[x% of 9 gallons] + [48% of 8 gallons] = [30% of 17 gallons]
9x+48(8)=30(17)
9x+384=510
9x=510-384
9x=126
x=126/9
x=14
Hence final answer is D) 14%
Answer:D
Step-by-step explanation:jus took test
Which of the following is an even function?
Answers
Answer:
I believe it is the third one (C)
Step-by-step explanation:
Hello There!
Your answer would be the first one.
f(x) = \x/
F of x has to equal f of -x
can anybody help me solve this please?
Answers
Calculate the volume of each shape then add them together.
4 x 32 x 10 = 1280
12 x 16 x 10 = 1920
Total volume = 1280 + 1920 = 3,200 cubic in.
Someone help me with this question
Answers
Hello There!
The answer would be 0.6
This is because you have to multiply the probability of each of these. First, I know that 0.5 is 1/2 in fraction form and that 0.3 is 3/10
Is this a function, explain answer
Answers
No. There are two spots in the vertical line. (Vertical line test)
ANSWER
No, there are two points with the same x-coordinates.
EXPLANATION
The relation in the graph is not a function.
There are two points in the relation that have the same x-coordinates.
These points are (1,1) and (1,3).
As a result of this , a vertical line will intersect the graph of the relation at these two points.
Since the graph of the relation does not pass the vertical line test, it is not a function.
kayla is sending invitations to 20 of her freinds for a pizza party. if there are 12 invitatons left to end out, how many invitations has she already mailed?
Answers
Kayla has mailed out 8 invitations
Need help fast do not understand this one. Solve: (x+5) / (x+8)=1+(6) / (x+1) showing all work.
Answers
Answer: [tex]x=-\frac{17}{3}[/tex]
Step-by-step explanation:
Given the equation [tex]\frac{(x+5)}{(x+8)}=1+\frac{6}{(x+1)}[/tex], you need to make the addtition indicated on the right side of the equation:
[tex]\frac{(x+5)}{(x+8)}=\frac{(x+1)+6}{(x+1)}\\\\\frac{(x+5)}{(x+8)}=\frac{(x+7)}{(x+1)}[/tex]
Now, multiply both sides of the equation by (x+8) and (x+1):
[tex](x+1)(x+8)\frac{(x+5)}{(x+8)}=\frac{(x+7)}{(x+1)}(x+1)(x+8)\\\\(x+1)(x+5)=(x+7)(x+8)[/tex]
Now, apply Distributive property:
[tex]x^2+5x+x+5=x^2+8x+7x+56[/tex]
Simplifying, you get:
[tex]6x+5=15x+56[/tex]
Subtract 5 and 15x from both sides:
[tex]6x+5-15x-5=15x+56-15x-5\\\\-9x=51[/tex]
Finally, divide both sidesby -9:
[tex]\frac{-9x}{-9}=\frac{51}{-9}\\\\x=-\frac{17}{3}[/tex]
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
Lloyd's Bakery sold one customer 9 dozen chocolate cookies and 8 dozen oatmeal cookies for $110. The bakery also sold another customer 9 dozen chocolate cookies and 5 dozen oatmeal cookies for $89. How much do the cookies cost?
A dozen chocolate cookies cost $___ and a dozen oatmeal cookies cost $___
Answers
To describe the situation, we can write a system of equations. The cost of a dozen chocolate cookies and a dozen oatmeal cookies can be found by solving the system of equations.
Explanation:To write a system of equations to describe the situation, let's assign variables to the unknown quantities. Let's say the cost of a dozen chocolate cookies is x dollars and the cost of a dozen oatmeal cookies is y dollars.
The first equation represents the sale of 9 dozen chocolate cookies and 8 dozen oatmeal cookies for $110. This can be written as:
9x + 8y = 110
The second equation represents the sale of 9 dozen chocolate cookies and 5 dozen oatmeal cookies for $89. This can be written as:
9x + 5y = 89
We now have a system of two equations with two variables. We can solve this system using any method, such as substitution or elimination, to find the values of x and y.
Once we find the values of x and y, we can determine the cost of a dozen chocolate cookies and a dozen oatmeal cookies, respectively.
find the mean of these values 100, 200, 300, 400
Answers
250
100 + 200 + 300 + 400 = 1000
1000/4 = 250
Answer:
250
Step-by-step explanation:
Use the pattern to answer the question. 27, 18, 12, 8. Which expression can be used to find the nth term of the pattern?
A. 27-6n
B. 27-6(n-1)
C. 27(2/3)^n
D. 27(2/3)^n-1
Answers
If you cross check with the answer , you’ll find option C to be the most appropriate option.
The difference between the first two terms is 9 and 9 is not a multiple of six, therefore first two are wrong.
Similarly, if you check option D for the first two terms , you get the second term as 27 which is false.
Therefore option C is the answer, also , it satisfies the series.
Final answer:
The correct expression to find the nth term of the given pattern is Option D: [tex]27(2/3)^{(n-1)}[/tex], as it accurately represents the geometric sequence of the provided numbers.
Explanation:
To find the nth term of the given pattern 27, 18, 12, 8, we will consider each option and see which one fits the sequence of numbers.
The first term (n=1) is 27.
The second term (n=2) is 18, which is 27 multiplied by ⅔ or 2/3.
The third term (n=3) is 12, which is the second term (18) multiplied by ⅔ or 2/3.
Similarly, the fourth term (n=4) is 8, which is the third term (12) multiplied by ⅔ or 2/3.
We can see that with each step, the term is being multiplied by a factor of ⅔ or 2/3. Therefore, an expression that uses multiplication by ⅔ repeatedly would likely represent the pattern.
Let's test the options:
Option A: 27-6n does not fit as it suggests a linear pattern with a constant subtraction, which does not match our sequence.
Option B: 27-6(n-1) also suggests a linear pattern and will not result in the given sequence when evaluated for n=1,2,3,4.
Option C: [tex]27(2/3)^n[/tex] represents a geometric sequence with a common ratio of ⅔, but for n=1, this expression gives us 27*⅔ which is not equal to 27.
Option D: [tex]27(2/3)^{(n-1)}[/tex] when evaluated for n=1 gives us 27*[tex](2/3)^{(1-1)}[/tex] = 27*1 = 27, which is the correct first term of our sequence. Applying this for subsequent terms also gives the correct sequence. Therefore, this is the correct expression for the nth term of the pattern.
Thus, the expression to find the nth term of the pattern is Option D: [tex]27(2/3)^{(n-1)}[/tex].
A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train?
A. 120 metres
B. 180 metres
C. 324 metres
D. 150 metres
Answers
Answer:
D. 150 metres
Step-by-step explanation:
speed = (60) times 5/18 = 50/3
lenght of the train = (speed x time)
lenght = 50/3 times (9)
= 150 m.
I need help with solving this problem. A picture is attached! Ty
Answers
[tex]\bf \stackrel{\stackrel{\textit{net income}}{\textit{revenue}}}{f(x)}=9x^2-54x-144\implies \stackrel{\textit{revenue}}{0}=9x^2-54x-144 \\\\\\ 0=9(x^2-6x-16)\implies 0=x^2-6x-16 \\\\\\ 0=(x+2)(x-8)\implies x= \begin{cases} -2\\ 8&\checkmark \end{cases}[/tex]
since the units sold can't be negative, we can't use -2.
jane lives 1 3/8 miles from the airport. She drove from her house to the airport and back 2 times. How far did Jane drive? Move a number or numbers into the blank to show the answer
Answers
Answer:
[tex]5\frac{1}{2}\ mi[/tex]
Step-by-step explanation:
we know that
[tex]1\frac{3}{8}\ mi=\frac{1*8+3}{8}=\frac{11}{8}\ mi[/tex]
She drove from her house to the airport and back 2 times
so
Multiply the distance from her house to the airport by 4
[tex](4)*\frac{11}{8}=\frac{11}{2}=5\frac{1}{2}\ mi[/tex]
Summer camp surveys 40 campus to see if they would take tennis next week. 12 camper say they would. If they were 250 campers. How many campers should the counselors plan on for next week’s tennis lesson
Answers
Answer:
76.8
Step-by-step explanation:
40/250 = 0.16 (1 campers vote)
0.16 x 12 = 1.92
1.92 x 40 = 76.8
To estimate the number of campers who might be interested in tennis lessons, a proportion can be used based on a survey of 40 campers, 12 of whom were interested. Scaling this up to the total 250 campers, approximately 75 campers might be interested in the tennis lessons.
The question is asking us to estimate the number of campers who would be interested in taking tennis lessons at a summer camp. From a small survey of 40 campers, 12 said they would take tennis lessons. To find out how many campers out of the total 250 might be interested, we need to use proportion. The proportion of campers interested in the survey is 12 out of 40, which we can set as equal to x out of 250, where x is the number we want to find. The equation is set up as follows:
12 / 40 = x / 250
Multiply both sides by 250 to isolate x:
250 × (12 / 40) = x
Calculate x:
x = 250 × (12 / 40) = 75
Therefore, the counselors should plan on approximately 75 campers attending next week’s tennis lesson.
9. The points (4, –6) and (9, -6) represent the location of two towns on a
coordinate grid, where one unit is equal to one mile. What is the distance, in miles, between the two towns?
Answers
Answer:
5 miles apart
Step-by-step explanation:
(4,-6) is 5 units away from (9,-6)
Mike spent $7920 on his vacation, which was 11% of his monthly salary. What was his monthly salary?
Answers
Answer : 72,000
Ex: Let x be his monthly salary.
The equation to set up would be
7920=0.11x
This equation represents 11% of monthly salary = 7920
To solve you divide 0.11x by 0.11 to get x by itself. then you do what you do to one side to the other. therefore you divide 7920 by 0.11 and you get 72,000
Plz help me with this
Answers
Answer: C) the variance
Step-by-step explanation:
Variance is the "average" of the squared differences from the Mean.
Which number line represents the solution to |-2x|=4
Answers
Step-by-step explanation:
l - 2 x l = 4
2 x = 4
x = 2
Answer:
Answer:
-2 & 2
( option C )
find the value of n such that b^2 + 16b + n is a perfect square trinomial
Answers
Answer:
64
Step-by-step explanation:
To complete the square
add ( half the coefficient of the b- term )² to b² + 16b
b² + 2(8)b + 8²
= b² + 16b + 64 = (b + 8)²
Leila has $360 in twenty-dollar bills. How many twenty-dollar bills does she have?
Answers
18 $20 Dollar Bills
5 Bills In 100
10 Bills In 200
15 Bills In 300
18 Bills In 360
andre drinks 15 ounces of water, which is 3/5 of a bottle. How much does the bottle hold? Use x for the number of ounces of water the bottle holds.
Answers
Answer:
25 ounces of water
Step-by-step explanation:
3/5 = 15
Cross multiplying this:
1 = x
3/5 = 15
3/5x=15
x=25
Answer:
The bottle hold 25 ounces of water
Step-by-step explanation:
To find the amount of water the bottle hold, we will follow the steps below;
let x be the number of ounces of water the bottle holds
3/5 of x = 15 ounces
3/5 × x = 15
[tex]\frac{3X}{5}[/tex] = 15
Multiply both-side of the equation by 5
[tex]\frac{3X}{5}[/tex] × 5 = 15×5
At the left-hand side of the equation 5 will cancel-out 5 leaving us with 3x
3x = 75
Divide both-side of the equation by 3
[tex]\frac{3X}{3}[/tex] = [tex]\frac{75}{3}[/tex]
(On the left-hand side of the equation 3 at the numerator will cancel-out 3 at the denominator leaving us with x while on the right-hand side of the equation 75 will be divided by 3)
x = 25 ounce
Which graph represents the compound inequality? n<-2 or n 24
Answers
3 is correct, because for n more than 4 we must have node on x=4 too
What is the equation !!! Math 3. 10 points HELP NEEDED!!!
Answers
ANSWER
B.
[tex]{f}^{ - 1} (x) = {x}^{2} - 3[/tex]
EXPLANATION
Given
[tex]f(x) = \sqrt{x + 3} [/tex]
Let
[tex]y= \sqrt{x + 3} [/tex]
Interchange x and y.
[tex]x= \sqrt{y + 3} [/tex]
Square both sides
[tex] {x}^{2} = y + 3[/tex]
Solve for y
[tex]y = {x}^{2} - 3[/tex]
Therefore the inverse of f(x) is
[tex] {f}^{ - 1} (x) = {x}^{2} - 3[/tex]
PLEASEE HEELPPP I NEED THIS QUESTION ANYONE HELP!!!
Answers
Answer:
4u= - 4i - 24j
u +w = -9i - 3j
Step-by-step explanation:
The question is on operation of vectors
Given u= -i -6j and w= -8i+3j
Then 4u = 4{ -i-6j}
4u= - 4i - 24j
u+w= -i -6j + -8i+3j......................collect like terms
= -i -8i + -6j +3j
= -9i + -3j
= -9i - 3j
select the equation of the line that passes through the point (3,-1) and is parallel to the line on the graph
Answers
ANSWER
y=-1
EXPLANATION
The given line in the graph has equation
[tex]y = 2.5[/tex]
This line is a horizontal line.
All horizontal line has a slope of 0.
Let the equation of the line that is parallel to this line be
[tex]y = mx + b[/tex]
The slope of this line must also be zero.
We substitute the slope, m=0 and the point (3,-1) into this equation to get:
[tex] - 1 = 0(3) + b[/tex]
[tex] \therefore \: b = 0[/tex]
Hence the required equation is
[tex]y = - 1[/tex]
ANSWER
y= -1
Step-by-step explanation:
The given line in the graph has equation
This line is a horizontal line.
All horizontal line has a slope of 0.
Let the equation of the line that is parallel to this line be
The slope of this line must also be zero.
We substitute the slope, m=0 and the point (3,-1) into this equation to get:
Hence the required equation is
Find the value of each variable
Answers
Answer:
x=6.5
Step-by-step explanation:
This question is on intersecting cords and segments that intersect outside the circle
Forming an equation for this relation;
6(x+6)= 5(5+10)
6x+36=5(15)
6x+36=75.............................collect like terms
6x=75-36
6x=39...........................divide both sides by 6
x=6.5
doroteo determined that the correlation coefficient for a set data was -0.9. what percent of the total variation in the y-values is not explained by the regression line?
Answers
Answer:
19% of the total variation in the y-values is not explained by the regression line
Step-by-step explanation:
The correlation coefficient, r, is a measure of the degree of association between two variables. It gives information on the direction and strength of the relationship.
On the other hand, [tex]R^{2}[/tex] is the coefficient of determination and is a measure of the percent of the total variation in the y-values explained by the regression line.
[tex]R^{2}[/tex] is simply the square of r;
[tex]R^{2}=r^{2}=(-0.9)^{2}=0.81[/tex]
This implies that 81% of the total variation in the y-values is explained by the regression line. Therefore, only 19% is not explained.