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(1) A circle with an area of
8pi square centimeters is dilated so that its image has an area of
32 pi square centimeters. What is the scale factor of the dilation?
A 2
B 4
C 8
D 16

( $1$ ) A circle with an area of $8 \pi$ square centimeters is dilated so that its image has an area of $32 \pi$ square centimeters. What is the scale factor of the dilation? $\newline$ A $2$ $\newline$ B $4$ $\newline$ C $8$ $\newline$ D $16$

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Solve quadratic equations: word problems

Full solution

Q. (

1

) A circle with an area of

8 \pi

square centimeters is dilated so that its image has an area of

32 \pi

square centimeters. What is the scale factor of the dilation?

\newline

A

2

\newline

B

4

\newline

C

8

\newline

D

16

Given Information: Original area of the circle is $8\pi$ square centimeters. New area of the circle is $32\pi$ square centimeters. The area of a circle is proportional to the square of its radius.
Define Scale Factor: Let's call the scale factor ' $k$ '. The new area is $k^2$ times the original area because when a figure is dilated, the area is multiplied by the square of the scale factor.
Set Up Equation: Set up the equation: $k^2 \times \text{original area} = \text{new area}$ . So, $k^2 \times 8\pi = 32\pi$ .
Solve for $k^2$ : Divide both sides by $8\pi$ to solve for $k^2$ : $k^2 = \frac{32\pi}{8\pi}$ .
Final Scale Factor: Simplify the equation: $k^2 = 4$ .
Final Scale Factor: Simplify the equation: $k^2 = 4$ . Take the square root of both sides to solve for $k$ : $k = 2$ or $k = -2$ . Since a scale factor cannot be negative, $k = 2$ .

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