Question

practice using scale to find areas
study the example showing how to find an actual
area from a scale drawing. then solve problems 1 - 4.
example
a camping supply store uses a tent as its logo.
the store makes a sign with the logo on it. the scale
from the drawing of the logo to the logo on the sign
is 1 in. to 2 ft. what is the area of the logo on the sign?
you can use a scale factor to find the
dimensions on the sign. the scale factor from
the logo to the sign is \\( \frac { 2 } { 1 } \\), or 2.
\\( \

$$\begin{aligned} a & = \\frac { 1 } { 2 } b h \\\\ & = \\frac { 1 } { 2 } ( 1.5 \\times 2 ) ( 1.5 \\times 2 ) \\\\ & = 4.5 \\end{aligned}$$

\\)
the area of the logo on the sign is \\( 4.5 \mathrm { ft } ^ { 2 } \\).
employees of the store in the example wear shirts with the logo on the back. the
scale from the original logo in inches to the logo on the shirt in inches is \\( 1 : 8 \\).
what is the area of the logo on the shirts? show your work.

Explanation:

Step1: Find the scale factor

The scale is \(1:8\), so the scale factor \(k = 8\).

Step2: Calculate the area of the original triangle

The original triangle has base \(b = 1.5\) in and height \(h=1.5\) in. Using the formula for the area of a triangle \(A=\frac{1}{2}bh\), we have \(A_{original}=\frac{1}{2}\times1.5\times1.5=\frac{1}{2}\times2.25 = 1.125\) \(in^{2}\).

Step3: Use the scale - area relationship

The ratio of the areas of two similar figures is \(A_{new}=A_{original}\times k^{2}\). Since \(k = 8\), then \(A_{new}=1.125\times8^{2}\).

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Answer:

The area of the logo on the shirts is \(72\) \(in^{2}\).