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3 A solid with surface area 8 square units is dilated by a scale factor of
k to obtain a solid with surface area
A square units. Find the value of
k which leads to an image with each given surface area.
PART A
512 square units
PART B

(1)/(2) square unit
PART C
8 square units

$3$ A solid with surface area $8$ square units is dilated by a scale factor of $k$ to obtain a solid with surface area $A$ square units. Find the value of $k$ which leads to an image with each given surface area. $\newline$ PART A $\newline$ $512$ square units $\newline$ PART B $\newline$ $\frac{1}{2}$ square unit $\newline$ PART C $\newline$ $8$ square units

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Scale drawings: word problems

Full solution

Q.

3

A solid with surface area

8

square units is dilated by a scale factor of

k

to obtain a solid with surface area

A

square units. Find the value of

k

which leads to an image with each given surface area.

\newline

PART A

\newline

512

square units

\newline

PART B

\newline

\frac{1}{2}

square unit

\newline

PART C

\newline

8

square units

Find Scale Factor for PART A: Find the scale factor $k$ for PART A where the surface area becomes $512$ square units. $\newline$ Surface area after dilation = $k^2 \times \text{original surface area}$ . $\newline$ $512 = k^2 \times 8$ . $\newline$ $k^2 = \frac{512}{8}$ . $\newline$ $k^2 = 64$ . $\newline$ $k = \sqrt{64}$ . $\newline$ $k = 8$ .
Find Scale Factor for PART B: Find the scale factor $k$ for PART B where the surface area becomes $\frac{1}{2}$ square unit. $\newline$ Surface area after dilation = $k^2 \times \text{original surface area}$ . $\newline$ $\frac{1}{2} = k^2 \times 8$ . $\newline$ $k^2 = \frac{\frac{1}{2}}{8}$ . $\newline$ $k^2 = \frac{1}{16}$ . $\newline$ $k = \sqrt{\frac{1}{16}}$ . $\newline$ $k = \frac{1}{4}$ .
Find Scale Factor for PART C: Find the scale factor $k$ for PART C where the surface area remains $8$ square units. $\newline$ Surface area after dilation = $k^2 \times \text{original surface area}$ . $\newline$ $8 = k^2 \times 8$ . $\newline$ $k^2 = \frac{8}{8}$ . $\newline$ $k^2 = 1$ . $\newline$ $k = \sqrt{1}$ . $\newline$ $k = 1$ .

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