An architect’s plan uses the scale . A photocopy of the plan enlarges all lengths by a factor of . On the photocopy, a rectangular room measures centimeters by centimeters. Which choice is closest to the area, in square feet, of the actual room? (Use .)

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Super-Hard SAT Math Question 22 | Free SAT Prep | Aniko

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Question 22·200 Super-Hard SAT Math Questions·Problem Solving and Data Analysis

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An architect’s plan uses the scale $1\text{ cm}=0.5\text{ m}$ . A photocopy of the plan enlarges all lengths by a factor of $1.5$ . On the photocopy, a rectangular room measures $10.8$ centimeters by $6.75$ centimeters.

Which choice is closest to the area, in square feet, of the actual room? (Use $1\text{ m}=3.28\text{ ft}$ .)

When a diagram is resized (like a photocopy), undo that resize first (divide by the enlargement factor). Then apply the drawing’s scale to convert lengths to real units, compute area in real units, and finally convert area units by squaring any linear conversion factor (because area is two-dimensional).

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Account for the photocopy

The photocopy enlarges lengths by $1.5$ . To get the original plan lengths, reverse that change.

Apply the plan scale

After you have the original plan dimensions (in cm), use $1\text{ cm}=0.5\text{ m}$ to convert each side to meters.

Compute area in $\text{m}^2$ first

Multiply the real-life length and width (in meters) to get the area in $\text{m}^2$ .

Square the meter-to-foot factor

To convert $\text{m}^2$ to $\text{ft}^2$ , you must square the linear conversion factor $3.28$ .

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Recover the original plan dimensions

Enter 10.8/1.5 and 6.75/1.5 to get 7.2 and 4.5 (in cm).

Convert to real lengths (meters) and find area

Enter 7.2*0.5 and 4.5*0.5 to get 3.6 and 2.25, then enter 3.6*2.25 to get 8.1 (in square meters).

Convert to square feet

Enter 8.1*(3.28)^2 to get about 87.1, then pick the closest option.

Step-by-step Explanation

Undo the photocopy enlargement

The photocopy enlarges lengths by a factor of $1.5$ , so divide each photocopy measurement by $1.5$ to get the original plan measurements:

$10.8\div 1.5=7.2$ cm
$6.75\div 1.5=4.5$ cm

Use the plan scale to get real dimensions (meters)

Using $1\text{ cm}=0.5\text{ m}$ :

Length: $7.2(0.5)=3.6$ m
Width: $4.5(0.5)=2.25$ m

Find the real area in square meters

A=3.6\times 2.25=8.1

So the area is $8.1\text{ m}^2$ .

Convert $\text{m}^2$ to $\text{ft}^2$ and choose the closest option

Since $1\text{ m}=3.28\text{ ft}$ , then

1\text{ m}^2=(3.28)^2\text{ ft}^2

8.1(3.28)^2\approx 8.1(10.7584)\approx 87.1

The closest choice is $87$ .

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Step-by-step Explanation

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