Question

the shadow of a vertical tower is 70.0 ft long when the angle of elevation of the sun is 34.0°. find the height of the tower.

the tower is □ ft tall.
(simplify your answer. type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Identify the trigonometric relationship

We can model this situation as a right triangle, where the height of the tower is the opposite side (\(h\)) to the angle of elevation (\(\theta = 34.0^\circ\)), and the length of the shadow is the adjacent side (\(x = 70.0\) ft). The tangent function relates the opposite and adjacent sides in a right triangle: \(\tan(\theta)=\frac{h}{x}\).

Step2: Solve for the height \(h\)

Rearranging the formula for \(h\), we get \(h = x\times\tan(\theta)\). Substituting \(x = 70.0\) ft and \(\theta = 34.0^\circ\) into the formula, we have \(h = 70.0\times\tan(34.0^\circ)\). Using a calculator to find \(\tan(34.0^\circ)\approx0.6745\), then \(h = 70.0\times0.6745 = 47.215\). Rounding to the nearest tenth, we get \(h\approx47.2\).

Answer:

47.2