Question

john and marsha are calculating the width of the lake shown in the figure. theyre preparing a report for school and want to create a model of the lake in the report. if they make the length of the short leg of the triangle in the figure 3 inches long, how long will they make the long leg?
a) 5.2 inches
b) 5.3 inches
c) 5.6 inches
d) 5.1 inches

Explanation:

Step1: Identify the triangle type and ratio

The triangle in the figure is a right - triangle with one angle of \(60^{\circ}\). In a \(30 - 60-90\) triangle, the ratio of the short leg (opposite \(30^{\circ}\)) to the long leg (opposite \(60^{\circ}\)) is \(1:\sqrt{3}\approx1: 1.732\). But we can also use the given real - world lengths. The short leg in real life is \(90\) ft and the long leg is \(156\) ft. We can set up a proportion for the model. Let the length of the long leg in the model be \(x\) inches. The proportion is \(\frac{\text{Short leg in real life}}{\text{Short leg in model}}=\frac{\text{Long leg in real life}}{\text{Long leg in model}}\), that is \(\frac{90}{3}=\frac{156}{x}\) (we can also think in terms of the ratio of the sides of the triangle. The ratio of the long leg to the short leg in real life is \(\frac{156}{90}=\frac{26}{15}\approx1.733\)).

Step2: Solve the proportion

We know that the ratio of the long leg to the short leg in real life is \(\frac{156}{90}=\frac{26}{15}\). In the model, the short leg is \(3\) inches. Let the long leg be \(x\) inches. Then, using the proportion \(\frac{x}{3}=\frac{156}{90}\). Cross - multiply: \(90x = 3\times156\). So \(90x=468\). Then \(x=\frac{468}{90}=\frac{26}{5} = 5.2\) inches.

Answer:

A) 5.2 inches