Chapter 6.6, Problem 51E

a.

To determine

To calculate: The height of the actual house.

Answer to Problem 51E

Height of the actual house = $\text{36 feet}$

Explanation of Solution

Given information:

Scale = $\text{1 inch}:1\text{ yard}$

Height of the scale model = $\text{1 foot}$

Calculation:

To find height of the actual house, use the ratio of the corresponding sides and form a proportion. Let height of the actual house be $x$ yards.

  $\begin{array}{l}\frac{1\text{ inch}}{1\text{ yard}}=\frac{1\text{ foot}}{x}\text{ [Form a proportion}]\\ \frac{1\text{ inch}}{1\text{ yard}}=\frac{12\text{ inch}}{x}\text{ [Convert foot to inches, 1 foot =12inches]}\\ 1\cdot x=12\cdot 1\text{ [Form cross product]}\\ x=12\text{ [Multiply]}\\ \end{array}$

Therefore, height of the actual house = $\text{12 yards}=12\times 3=36\text{ feet}$

b.

To determine

To calculate: Area of the deck in the model.

Answer to Problem 51E

Area of model’s deck = $0.14\text{ square feet}$

Explanation of Solution

Given information:

The house’s deck is $\text{15 foot}$ by $\text{12 foot}$ rectangle.

Calculation:

To find area of deck in the model, use properties of similar figures. The ratio of corresponding sides of similar figures are equal. Let length of the model’s deck be $l$ feet and width be $w$ feet. Hence, form a proportion to find length and width.

For length:

  $\begin{array}{l}\frac{1\text{ inch}}{1\text{ yard}}=\frac{l\text{ ft}}{15\text{ ft}}\text{ [Form a proportion}]\\ \frac{1\text{ inch}}{\text{36 inches}}=\frac{l\text{ ft}}{15\text{ ft}}\text{ [Convert yard to inches, 1 yard =36 inches]}\\ 1\cdot 15=l\cdot 36\text{ [Form cross product]}\\ l=\frac{15}{36}\text{ [Divide both sides by 36]}\\ l=\frac{5}{12}\text{ ft [Express in simplest form]}\end{array}$

For width:

  $\begin{array}{l}\frac{1\text{ inch}}{1\text{ yard}}=\frac{w\text{ ft}}{12\text{ ft}}\text{ [Form a proportion}]\\ \frac{1\text{ inch}}{\text{36 inches}}=\frac{w\text{ ft}}{12\text{ ft}}\text{ [Convert yard to inches, 1 yard =36 inches]}\\ 1\cdot 12=w\cdot 36\text{ [Form cross product]}\\ w=\frac{12}{36}\text{ [Divide both sides by 36]}\\ w=\frac{1}{3}\text{ ft [Express in simplest form]}\end{array}$

Area of model’s deck

  $\begin{array}{l}\text{=Length}\times \text{width [Formula for area]}\\ =\frac{5}{12}\times \frac{1}{3}\text{ [Substitute] }\\ =\frac{5}{36}\text{ [Multiply]}\\ =0.14\text{ square feet [Write fraction as decimal]}\end{array}$