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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 $512$ square units $\newline$ PART B $\frac{(1)}{(2)}$ square unit $\newline$ PART C $8$ square units

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Scale drawings: word problems

Full solution

Q. 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

512

square units

\newline

PART B

\frac{(1)}{(2)}

square unit

\newline

PART C

8

square units

Calculate scale factor for PART A: Find the scale factor $k$ for PART A where the surface area after dilation is $512$ square units. $\newline$ Ratio of new surface area to original surface area = $A /$ original surface area. $\newline$ Ratio = $512 / 8$ . $\newline$ Calculate the ratio. $\newline$ Ratio = $64$ . $\newline$ Since the ratio of the areas is the square of the scale factor, take the square root to find $k$ . $\newline$ $k = \sqrt{64}$ . $\newline$ Calculate $k$ . $\newline$ $k = 8$ .
Calculate scale factor for PART B: Find the scale factor $k$ for PART B where the surface area after dilation is $(1)/(2)$ square units. $\newline$ Ratio of new surface area to original surface area = $A /$ original surface area. $\newline$ Ratio = $(1)/(2) / 8$ . $\newline$ Calculate the ratio. $\newline$ Ratio = $(1)/(16)$ . $\newline$ Since the ratio of the areas is the square of the scale factor, take the square root to find $k$ . $\newline$ $k = \sqrt{1/16}$ . $\newline$ Calculate $k$ . $\newline$ $k = 1/4$ .
Calculate scale factor for PART C: Find the scale factor $k$ for PART C where the surface area after dilation is $8$ square units. $\newline$ Ratio of new surface area to original surface area = $A /$ original surface area. $\newline$ Ratio = $8 / 8$ . $\newline$ Calculate the ratio. $\newline$ Ratio = $1$ . $\newline$ Since the ratio of the areas is the square of the scale factor, take the square root to find $k$ . $\newline$ $k = \sqrt{1}$ . $\newline$ Calculate $k$ . $\newline$ $k = 1$ .

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