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A rectangle with an area of 140 units
^(2) is the image of a rectangle that was dilated by a scale factor of
(3)/(4). Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary.
Answer: units
^(2)

A rectangle with an area of $140$ units $^{2}$ is the image of a rectangle that was dilated by a scale factor of $\frac{3}{4}$ . Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary. $\newline$ Answer: units $^{2}$

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Scale drawings: word problems

Full solution

Q. A rectangle with an area of

140

units

^{2}

is the image of a rectangle that was dilated by a scale factor of

\frac{3}{4}

. Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary.

\newline

Answer: units

^{2}

Understand problem and relationship: Understand the problem and the relationship between the area of the image and the preimage when dilation occurs. $\newline$ The area of the image after dilation is given as $140$ square units. The scale factor of the dilation is $\frac{3}{4}$ , which means every linear dimension of the preimage is multiplied by $\frac{3}{4}$ to get the image. The area of the preimage can be found by dividing the area of the image by the square of the scale factor.
Calculate preimage area: Calculate the area of the preimage using the scale factor. $\newline$ The area of the preimage $A_{\text{preimage}}$ is equal to the area of the image $A_{\text{image}}$ divided by the square of the scale factor $k^2$ . $\newline$ $A_{\text{preimage}} = \frac{A_{\text{image}}}{k^2}$ $\newline$ $A_{\text{preimage}} = \frac{140}{(\frac{3}{4})^2}$
Perform calculation: Perform the calculation. $\newline$ $A_{\text{preimage}} = \frac{140}{\left(\frac{9}{16}\right)}$ $\newline$ To divide by a fraction, multiply by its reciprocal. $\newline$ $A_{\text{preimage}} = 140 \times \left(\frac{16}{9}\right)$ $\newline$ $A_{\text{preimage}} = 140 \times \frac{16}{9}$ $\newline$ $A_{\text{preimage}} = \frac{2240}{9}$ $\newline$ $A_{\text{preimage}} \approx 248.9$ (rounded to the nearest tenth)

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