Question

Jessica wants to plan where she will place furniture in her room, so she has drawn a scale model of her room in order to begin planning. the scale model is 11 inches long and 8.5 inches wide. jessica notices that the door to her room takes up 1.75 inches on the right-side wall of her room, and she realizes that she cannot place any furniture in the area that the door turns through when it is opened. if the longest wall in jessica’s actual room is 15 feet long, what is the best estimate of the area of the actual space in her room where furniture cannot be placed because of the door? 6 ft 2 6.5 ft 2 18 ft 2 9 ft 2

Answer 1

(d) 9 ft²

Explanation:

To find the excluded area for furniture, we can find the actual size of the door, then compute the area of the semicircle that it turns through. The door size will be the product of the drawing dimension and the scale factor relating the room to the drawing.

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door size

Room size and drawing size are proportional. This means ...

door size / door drawing = room length / room drawing

door size / (1.75 in) = (15 ft)/(11 in)

door size = (15 ft)(1.75/11) = 105/44 ft ≈ 2.386 ft

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exclusion area

The area of a circle with radius r is given by the formula ...

A = πr²

Then the area of a semicircle (half a circle) with radius 2.386 ft will be ...

A = 1/2π(2.386 ft)² ≈ 8.945 ft²

A good estimate of the furniture exclusion area is 9 ft².

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Additional comment

The scale factor is (15 ft)/(11 in) = 15/11 ft/in. Multiplying that by 7/4 in gives 105/44 ft.