Question

a tree on a hillside casts a shadow 210 ft down the hill, as shown in the following figure.
if the angle of inclination of the hillside is 21° to the horizontal and the angle of elevation of the sun is 51°, find the height of the tree. (round your answer to the nearest foot.)

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326
× ft

Explanation:

Step1: Find the angle between the hillside and the sun's ray

The angle between the hillside and the sun's ray is \(51^{\circ}- 21^{\circ}=30^{\circ}\)

Step2: Use the sine law

Let the height of the tree be \(h\). According to the sine law \(\frac{h}{\sin30^{\circ}}=\frac{210}{\sin(90^{\circ}-51^{\circ})}\)

Since \(\sin(90^{\circ}-51^{\circ})=\cos51^{\circ}\approx0.6293\) and \(\sin30^{\circ} = 0.5\)

We have \(h=\frac{210\times\sin30^{\circ}}{\cos51^{\circ}}\)

Step3: Calculate the value of \(h\)

Substitute the values: \(h=\frac{210\times0.5}{0.6293}=\frac{105}{0.6293}\approx167\)

Answer:

\(167\)