There are 120 ways we can list 5 basketball players in order in a program.
What are permutations?Permutations are the selection in order from a given set of values. Here, the order is of importance.
We can find the number of ways of listing the basketball players below:There are five basketball players.
We are asked to find the number of ways in which they can be listed in order.
We can use permutations to find the answer.
We can find the number of ways in which the basketball players can be listed in order as follows:
Total number of ways = 5*4*3*2*1
= 20*3*2*1
= 60*2*1
= 120 ways.
The number of ways in which the basketball players can be listed in order is found. The basketball ball can be listed in 120 ways.
Therefore, we have found that there are 120 ways we can list 5 basketball players in order in a program.
Learn more about permutations here: https://brainly.com/question/1216161
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If m+p+k=70, p=2m, and k=2p, then m =
m=10 as p=2m and k=2p=4m
Answer:
10
Step-by-step explanation:
which statement best describes the association between variable X and variable Y?
*weak negative association
*moderate negative association
*moderate positive association
*weak positive association
Answer:
Weak negative association
Step-by-step explanation:
because it is widely scattered, it is weak but in general it is negative
Answer:
weak negative association
Step-by-step explanation:
The association is negative because when variable x increase, variable y decrease.
Given that many points, like x = 1, x = 2, et cetera, have more than 1 y-value associated and those values are spread, the association is weak.
There were 15 girls and 25 boys in the Science club. What percent of the members were girls?
Answer: 37.5
Step-by-step explanation:
15 15 3
___ = ___ = ___ = 0.375 = 37.5
15+25 40 8
The Pythagorean Theorem says that in a right triangle, a2 + b2 = c2 , where a and b are the lengths of the legs, and c is the length of the hypotenuse. If b = 2, what is the length of side a in terms of c? Remember, “in terms of c” means that the variable c will appear in your answer.
Answer: [tex]a=\sqrt{c^2-4}[/tex]
Step-by-step explanation:
You know that the Pythagorean Theorem is:
[tex]a^2+b^2=c^2[/tex]
Where "a" and "b" are the legs and "c" is the hypotenuse.
Then, since you need to find the length of side "a" in terms of the hypotenuse "c", you need to solve for "a":
Subtract b² from both sides of the equation:
[tex]a^2+b^2-b^2=c^2-b^2[/tex]
[tex]a^2=c^2-b^2[/tex]
And finally, you need to apply square root to both sides of the equation:
[tex]\sqrt{a^2}=\sqrt{c^2-b^2}\\\\a=\sqrt{c^2-b^2}[/tex]
Then:
[tex]a=\sqrt{c^2-2^2}\\\\a=\sqrt{c^2-4}[/tex]
Answer:
Final answer is [tex]a=\sqrt{c^2-4}[/tex].
Step-by-step explanation:
Given that b=2. Now using Pythagorean theorem, we need to find the value of a in terms of c.
So let's plug b=2 into formula :
[tex]a^2+b^2=c^2[/tex]
[tex]a^2+2^2=c^2[/tex]
[tex]a^2+4=c^2[/tex]
[tex]a^2=c^2-4[/tex]
Take square root of both sides and use principle root as side length can't be negative.
[tex]a=\sqrt{c^2-4}[/tex]
Hence final answer is [tex]a=\sqrt{c^2-4}[/tex].
Michael has a recipe that requires 3/4 cup of flour,and he wants to make 1/2 the recipe. How much flour does he need?
Answer:
Michael will use 3/8 cup of flour.
Step-by-step explanation:
Half of a Fraction:
1. Reduce fraction to its lowest terms.
2. If the numerator is an even number than divide the numerator by 2.
Example: 4/5 divided in half would be 2/5.
3. If the numerator is an odd number than multiply the denominator by 2.
Example: 1/3 divided in half would be 1/6.
3/4 cup of flour divided in half would be 3/8 because the numerator 3 is an odd number so the denominator of 4 would be multiplied by 2 equaling 8.
solve using the substitution method 3m-n=18 and 2m+n=-7
Answer:
m = 11/5, n= -57/5
Step-by-step explanation:
3m-n=18 and 2m+n=-7
Solve one of the equations for n
2m +n = -7
Subtract 2m from each side
2m-2m +n = -7 -2m
n = -7-2m
Substitute this into the first equation
3m -n =18
3m - (-7-2m) = 18
Distribute the minus sign
3m +7+2m = 18
Combine like terms
5m +7 = 18
Subtract 7 from each side
5m+7-7 = 18-7
5m = 11
Divide by 5
m = 11/5
Substitue this back into the equation for n
n = -7-2m
=-7 -2(11/5)
=-7-22/5
-35/5 -22/5
=-57/5
To solve the system using substitution, solve the first equation for n, substitute it into the second, and solve for m, which gives m = 11/5. Then, substitute m back into the expression for n to get n = -57/5.
To solve the system of equations using the substitution method, you can solve one of the equations for one variable and then substitute that expression into the other equation. Let's start with the two given equations:
3m - n = 18
2m + n = -7
First, solve the first equation for n:
n = 3m - 18
Now substitute this expression for n into the second equation:
2m + (3m - 18) = -7
Combine like terms:
5m - 18 = -7
Add 18 to both sides:
5m = 11
Divide by 5:
m = 11/5
Next, substitute the value of m back into the expression for n:
n = 3(11/5) - 18
n = 33/5 - 90/5
n = -57/5
Therefore, the solution to the system using the substitution method is m = 11/5 and n = -57/5.
Between x = 2 and x = 3, which function has a larger average rate of change than f(x) = 2x has?
Answer:
x=2
Step-by-step explanation:
Answer:
C) h(x)=[tex]4^{x-1}[/tex]
Step-by-step explanation:
10 points help please!?
Answer:
[tex]\large\boxed{\sqrt{80}=4\sqrt5}[/tex]
Step-by-step explanation:
Method 1:
The formula of a distance between two points:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
We have the points (-4, 1) and (4, 5). Substitute:
[tex]d=\sqrt{(4-(-4))^2+(5-1)^2}=\sqrt{8^2+4^2}=\sqrt{64+16}=\sqrt{80}[/tex]
[tex]\sqrt{80}=\sqrt{16\cdot5}=\sqrt{16}\cdot\sqrt5=4\sqrt5[/tex]
Method 2:
Look at the picture.
Use the Pythagorean theorem:
[tex]leg^2+leg^2=hypotenuse^2[/tex]
We have
[tex]leg=8,\ leg=4,\ hypotenuse=x[/tex]
Substitute:
[tex]x^2=8^2+4^2\\\\x^2=64+16\\\\x^2=80\to x=\sqrt{80}\\\\x=4\sqrt5[/tex]
how does f(x)=9^x change over the interval from x=8 to x=10
Answer:
Step-by-step explanation:
9^8 to 9^10 involves a change (multiplication) of 9^8 by 9^2 (which is 81).
Over the given interval, f(x) increases by 81.
A cylinder has a radius of 8 meters and a height of 4 meters. .
Given : Radius of cylinder = 8 m
Height = 4 m
Volume of cylinder = πr²h cu. units
= 22/7 × 8 × 8 × 4 m³
= 804.57 m³ (approx.)
Curved surface area = 2πrh sq. units
= 2 × 22/7 × 8 × 4 m²
= 201.14 m² (approx.)
Total surface area = 2πr(r + h) sq. units
= 2 × 22/7 × 8 (8 + 4) m²
= 2 × 22/7 × 8 × 12 m²
= 603.43 (approx.)
Final answer:
The volume of a cylinder with a radius of 8 meters and a height of 4 meters is calculated using the formula V = πr²h, which yields approximately 804.25 cubic meters when using π ≈ 3.14159.
Explanation:
To calculate the volume of a cylinder with a given radius and height, you can use the formula V = πr²h. In the case of a cylinder with a radius of 8 meters and a height of 4 meters, plug these values into the formula to get:
V = π × (8 m)² × 4 m
V = π × 64 m² × 4 m
V = 256π m³
Since π (pi) is approximately 3.14159, you can further calculate the volume as:
V ≈ 256 × 3.14159 m³
V ≈ 804.2477 m³
Thus, the volume of the cylinder is approximately 804.25 cubic meters, when expressed with four significant figures.
what is value of x-(3x+5) when x=-2
Answer:
2
Step-by-step explanation:
3 * -2 = -6
-6 +5 = 1
-2 * -1 = 2
the value would be two
a tree casts a shadow that is 20 feet long. if the tree is 35 feet tall, what is the angle of elevation from the end of the shadow to the top of the tree? round to the nearest tenth
Answer:
The angle of elevation is 60.3°
Step-by-step explanation:
* Lets revise the trigonometry functions
- In any right angle triangle:
# The side opposite to the right angle is called the hypotenuse
# The other two sides are called the legs of the right angle
* If the name of the triangle is ABC, where B is the right angle
∴ The hypotenuse is AC
∴ AB and BC are the legs of the right angle
- ∠A and ∠C are two acute angles
- For angle A
# sin(A) = opposite/hypotenuse
∵ The opposite to ∠A is BC
∵ The hypotenuse is AC
∴ sin(A) = BC/AC
# cos(A) = adjacent/hypotenuse
∵ The adjacent to ∠A is AB
∵ The hypotenuse is AC
∴ cos(A) = AB/AC
# tan(A) = opposite/adjacent
∵ The opposite to ∠A is BC
∵ The adjacent to ∠A is AB
∴ tan(A) = BC/AB
* Now lets solve the problem
∵ The shadow of the tree is 20 feet long
- The shadow of the tree is on the ground
∵ The height of the tree is 35 feet tall
∴ The shadow of the tree and the height of the tree formed the legs of
a right triangle
- The angle of elevation is opposite to the tree
∴ The shadow of the tree is the adjacent side of the angle of elevation
∴ The height of the tree is the opposite side of the angle of elevation
- let the name of the angle of elevation is Ф
∴ tan Ф = tree height/shadow length
∴ tan Ф = 35/20 = 7/4
∴ Ф = tan^-1(7/4) = 60.3°
* The angle of elevation is 60.3°
Solve for x
3x+3/x-1=3x+2/x+4
Answer:
X=3.5
Step-by-step explanation:
( − 2 7 ) ( 5 − 8 ) Please solve .
Answer:
81
Step-by-step explanation:
We have to solve the following multiplication: (−27) (5−8)
First, we are going to simplify the second parenthesis, before performing the multiplication:
(−27) (5−8) = (−27)(-3) = 81.
The result is positive given that minus times minus equals plus.
Step 1: Solve the second parentheses
( − 2 7 ) ( 5 − 8 )
5 - 8 = -3
(-27)(-3)
Step 2: Multiply -27 by -3. Since these are both negative the answer will be positive
-27 * -3 = 81
Hope this helped!
~Just a girl in love with Shawn Mendes
Find the value of y (Angle A and angle B are congruent)
A) 64
B) 68
C) 71
D) 82
Answer:
The value of y is 64 ⇒ first answer
Step-by-step explanation:
* Lets study the figure to solve the question
- The figure is a polygon of 5 sides
- It has five interior angles and five exterior angles
- The sum of its interior angles depends on the number of its sides
- We can find the sum of the measures of its interior angles from this
rule ⇒ the sum = (n - 2) × 180°, where n is the number of its sides
- The sum of the measures of its exterior angles is 360°
(fixed for any polygon)
- The sum of the measure of an interior angle and its exterior angle
is 180°
∵ m∠A = m∠B
∴ The exterior angle of ∠A = the exterior angle of ∠B
∵ The exterior angle of ∠B is y°
∴ The exterior angle of ∠A is y°
∵ The measure of the interior angle of the exterior angle x° is 90°
∴ 90° + x° = 180° ⇒ subtract 90 from both sides
∴ x° = 90°
∵ The polygon has five exterior angles
# Angle of measure 75 , angle of measure 67 , y° , y° , x°
∴ 75° + 67° + y° + y° + x° = 360° ⇒ sum of the exterior angles
∵ x° = 90°
∴ 75° + 67° + y° + y° + 90° = 360° ⇒ simplify
∴ 232 + 2y° = 360° ⇒ subtract 232 from both sides
∴ 2y° = 128 ⇒ divide both sides by 2
∴ y° = 64°
* The value of y is 64
Answer:
The correct answer is option A. 64
Step-by-step explanation:
From the figure we can see a pentagon.
Sum of angles of a pentagon is 540
To find the value of m<B
From the figure we get, Angle A and angle B are congruent
m<A = m<B and one angle is 90°
Other two angles are,
180 - 75 = 105° and 180 - 67 = 113°
Also we can write,
105 + 113 + 90 + m<A + m<B = 540
308 + m<A + m<B = 540
m<A + m<B = 540 - 308 = 232
2m<B = 232
m<B = 232/2 = 116
To find the value of y
From figure we get,
<B + y = 180
y = 180 - <B
= 180 - 116 = 64
Therefore the correct answer is option A. 64
If f(x)=3x2-2 and g(x)=4x+2 what is value of (f+g)(2)
The value of (f+g)(2) is 20.
The expression (f+g)(2) represents the sum of the functions f(x) and g(x) evaluated at x = 2.
To find the value of (f+g)(2), we need to first find the values of f(2) and g(2), and then add them together.
Given that f(x) = 3x^2 - 2 and g(x) = 4x + 2, we can find the values of f(2) and g(2) as follows:
1. Substitute x = 2 into f(x):
f(2) = 3(2)^2 - 2
= 3(4) - 2
= 12 - 2
= 10
2. Substitute x = 2 into g(x):
g(2) = 4(2) + 2
= 8 + 2
= 10
Now, we can find the value of (f+g)(2) by adding f(2) and g(2):
(f+g)(2) = f(2) + g(2)
= 10 + 10
= 20
Therefore, the value of (f+g)(2) is 20.
PLSSS HURRY!!!!!!!!
Solve the quadratic equation.
(x + 3)2 = 64
A) x = 5 or -11
B) x = 11 or -5
C) x = ± 3
D) x = ± 11
Answer:
A) x=5 or -11
Step-by-step explanation:
The given equation is:
[tex](x+3)^2=64[/tex]
The most suitable method so solve this quadratic equation is the square root method.
We take square root of both sides to obtain:
[tex]x+3=\pm \sqrt{64}[/tex]
[tex]\implies x+3=\pm8[/tex]
[tex]\implies x=-3\pm8[/tex]
We now split the plus or minus sign to get;
[tex]x=-3-8\:,x=-3+8[/tex]
This simplifies to:
[tex]x=-11\:,x=5[/tex]
The correct choice is A
15 points!!! The triangle ABC is dilated with respect to the point O(8,-3) and the scale factor 3 to a new triangle A'B'C'. What are the coordinates of B?
(24,-9)
(24,-3)
(12,-9)
(-4,-3)
Answer:
(-4,-3)
Step-by-step explanation:
we know that
The distance OB is equal to
(8-4)=4 units
To find the new distance OB', multiply the distance OB by the scale factor
so
OB'=OB*3
OB'=4*3=12 units
The x-coordinate of point B' is equal to
8-12=-4
The y-coordinate of point B' is the same that B -3
the coordinates of point B'are (-4,-3)
Can someone help me with this
Answer:
D) 3/25
Step-by-step explanation:
(3/5)/5 = .12
3/23 = .12
Hello There!
Your answer would be 3/25. Whenever you have two probabilities together you can multiply them and in this case, multiplying 3/5 by 1/5 will get you a product of 3/25
One of the same side angles of two parallel lines is 20° smaller than the other one. Find the measures of these two angles.
Answer:
The measure of these two angles are 80° and 100°
Step-by-step explanation:
Let
x and y ----> the measure of the same side angles
we know that
The sum of of the same side angles of two parallel lines is equal to 180 degrees
x+y=180° -----> equation A
x=y-20° ----> equation B
substitute equation B in equation A and solve for y
(y-20°)+y=180°
2y=180°+20°
y=200°/2
y=100°
Find the value of x
x=100°-20°=80°
A bacteria culture begins with 4 bacteria which double in size every hour. How many bacteria exist in the culture after 8 hours.
Answer:
1024
Step-by-step explanation:
There are 4 bacteria at the start. We can make an equation to represent the phenomena explained in the question.
As it is written in the question that the bacteria culture doubles in every hour.
So,
Let t represent the unit of time
So the number of bacteria after t unit of time will be
Number of bacteria after t unit of time=4*2^t
We have to calculate number of bacteria after 8 hours, so t = 8
Number of bacteria after 8 hours=4*2^8
=4*256
=1024
So the bacteria after 8 hours in the culture will be 1024..
The initial population of the bacterium cells was 4, and after 8 hours, there would be 1024 bacteria in the culture.
The initial population of the bacterium cells at the beginning of the experiment was 4 bacteria.
Step-by-step calculation:
After 1 hour: 4 * 2 = 8 bacteriaAfter 2 hours: 8 * 2 = 16 bacteriaContinuing this pattern, after 8 hours: 4 * (2⁸) = 4 * 256 = 1024 bacteria.Find the volume of the cylinder in terms of pi.
Answer:
[tex]\large\boxed{V=81\pi\ in^3}[/tex]
Step-by-step explanation:
The formula of a volume of a cylinder:
[tex]V=\pi r^2H[/tex]
r - radius
H - height
We have r = 3in and H = 9in. Substitue:
[tex]V=\pi(3^2)(9)=\pi(9)(9)=81\pi\ in^3[/tex]
Which interpretation for the given expression is correct? 5(3x-4)2 which one is right
A.
the quotient of 5 and the square of 3x - 4
B.
the product of 5 and the difference of 3x and the square of 4
C.
the product of 5 and the square of 3x - 4
D.
the difference of 5, 3x, and 4 squared
The correct interpretation for the expression 5(3x - 4)² is The product of 5 and the square of 3x - 4 ( option C)
An expression is a mathematical phrase containing variables, constants, and operators.
It represents a mathematical relationship or computation and does not have a specific value until the variables are assigned values.
Interpreting 5(3x - 4)²
is The product of 5 and the square of 3x - 4
Therefore, the correct interpretation for the expression 5(3x - 4)² is The product of 5 and the square of 3x - 4
1/3 2/5 4/7 least to greatest
1.Change them all to having common denominators so it’s easier to compare the numerators
1/3=35/105 *35
2/5=42/105 *21
4/7=60/105 *15
2.Compare and order
3. Answer from smallest to largest =
1/3,2/5,4/7
Hope this helps :)
-p+60 = = h + 10,000
In the equation above, h is a constant. If p = 10 is a solution to the equation, what is the value
of h?
By substituting p = 10 into the equation -p + 60 = h + 10,000 and solving for h, we find that the constant h is -9,950.
The solution cam be solved as:
To find the value of constant h when p = 10 is a solution to the equation -p + 60 = h + 10,000, we substitute p = 10 into the equation and solve for h.
-p + 60 = h + 10,000
-10 + 60 = h + 10,000
50 = h + 10,000
h = 50 - 10,000
h = -9,950
The value of h is therefore -9,950.
The perimeter of the rectangle is 28 units.
What is the value of w?
Step-by-step explanation:
The formula of a perimeter of a rectangle:
[tex]P=2l+2w[/tex]
We have P = 28. Substitute:
[tex]28=2l+2w[/tex]
Solve for w:
[tex]2l+2w=28[/tex] subtract 2l from both sides
[tex]2w=28-2l[/tex] divide both sides by 2
[tex]w=14-l[/tex]
Where [tex]0<l<14[/tex]
Answer:
answer is w = 14
25 Kaia ate 3/6 of a banana. Zoie ate an equivalent amount. Which
fraction shows how much of a banana Zoie ate? Circle the
letter of the correct answer.
A:1/3 B:2/3 C:5/8 or D:1/2
از
Landon chose A as the correct answer. How did he get
that answer?
Answer: D:1/2
Step-by-step explanation: 3/6 simplifies to 1/2, making them equal. Landon got his answer by simplifying the problem incorrectly. 1/3 is equal to 2/6
Final answer:
Kaia ate 3/6 of a banana, which simplifies to 1/2. Zoie ate an equivalent amount, so the fraction that shows this is 1/2, represented as option D:1/2. Landon incorrectly chose 1/3 which is not equivalent to 1/2.
Explanation:
Kaia ate 3/6 of a banana, which is an equivalent fraction to 1/2 when reduced (because 3/6 = 1/2). If Zoie ate an equivalent amount, then we are looking for a fraction that is also equivalent to 1/2. Among the options provided: A:1/3, B:2/3, C:5/8, and D:1/2, the correct answer is D:1/2. Landon chose A:1/3 as the correct answer, which is incorrect because 1/3 is not equivalent to 1/2.
To understand why 3/6 is equivalent to 1/2, we can divide the numerator and the denominator by their greatest common factor, which in this case is 3. Doing so, we simplify 3/6 as: 3÷3 / 6÷3 = 1/2, thus, showing that 3/6 and 1/2 represent the same quantity.
The histogram shows the weekly attendance of participants in a school's study skills program. What was the highest weekly attendance recorded for the program?
A.
8
B.
12
C.
16
D.
18
That would be on the 5th week. There were 18 students that attended (D.)
Hope this helped!
Answer: It is 18
Step-by-step explanation: Hope this helps :D
A pair of equations is shown below:
y = 6x − 4
y = 5x − 3
Part A: In your own words, explain how you can solve the pair of equations graphically. Write the slope and y-intercept for each equation that you will plot on the graph to solve the equations. (6 points)
Part B: What is the solution to the pair of equations? (4 points)
Answer:
The solution to the pair of equations is [tex]x=1,y=2[/tex]
Step-by-step explanation:
The given equations are:
[tex]y=6x-4[/tex]
[tex]y=5x-3[/tex]
To solve the pair of equations graphically, we need to graph the two equations. Their point of intersection is the solution to the pair of equations.
The functions are in the form;
[tex]y=mx+b[/tex]
where m=6 is the slope and b=-4 is the y-intercept of [tex]y=6x-4[/tex].
and where m=5 is the slope and b=-3 is the y-intercept of [tex]y=5x-3[/tex].
The two equations have been graphed in the attachment.
They intersected at (1,2).
The solution to the pair of equations is [tex]x=1,y=2[/tex]
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.
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: