Question

a city regulates the heights of its houses by the rule that on the shortest day of the year, a house cannot cast a noontime shadow that reaches farther than the shadow that would be cast by a 12 - foot fence on the property line. assume a 12 - foot fence casts a 10 - foot shadow on the shortest day of the year, and the house is set back 30 feet from the property line. the maximum allowed height for the house is ft. (simplify your answer. type an integer or a decimal.)

Explanation:

Step1: Set up proportion using similar - triangles

Since the triangles formed by the fence and its shadow and the house and its shadow are similar, the ratios of height to shadow length are equal. Let the height of the house be $h$. The ratio for the fence is $\frac{12}{10}$, and for the house, the maximum shadow length is $30 + 10=40$ feet (30 - foot set - back plus the 10 - foot shadow length of the fence). The proportion is $\frac{h}{40}=\frac{12}{10}$.

Step2: Solve the proportion for $h$

Cross - multiply the proportion: $10h = 12\times40$. Then $10h=480$. Divide both sides by 10: $h=\frac{480}{10}=48$.

Answer:

48