Question

from a boat on the lake, the angle of elevation to the top of a cliff is 10°35′. if the base of the cliff is 324 feet from the boat, how high is the cliff (to the nearest foot)?

a. 64 ft
b. 61 ft
c. 71 ft
d. 74 ft

Explanation:

Step1: Convert angle to decimal degrees

First, we need to convert the angle \(10^\circ35'\) to decimal degrees. Since \(1^\circ = 60'\), we have \(35'=\frac{35}{60}\approx0.5833^\circ\). So the angle in decimal degrees is \(10 + 0.5833=10.5833^\circ\).

Step2: Use tangent function

We can model this situation with a right triangle, where the adjacent side to the angle of elevation is the distance from the boat to the base of the cliff (\(324\) feet) and the opposite side is the height of the cliff (\(h\)) we want to find. The tangent of an angle in a right triangle is given by \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). So \(\tan(10.5833^\circ)=\frac{h}{324}\).

Step3: Solve for \(h\)

To find \(h\), we multiply both sides of the equation by \(324\): \(h = 324\times\tan(10.5833^\circ)\). Using a calculator to find \(\tan(10.5833^\circ)\approx0.186\) (more accurately, \(\tan(10.5833^\circ)\approx\tan(10^\circ35')\approx0.186\)). Then \(h = 324\times0.186\approx60.264\approx61\) (to the nearest foot). Wait, wait, maybe my approximation of the tangent was off. Let's calculate \(\tan(10^\circ35')\) more accurately. \(35\) minutes is \(35/60 = 7/12\approx0.583333\) degrees. So \(10.583333^\circ\). Using a calculator, \(\tan(10.583333^\circ)\): let's compute it. \(\tan(10^\circ)=0.1763\), \(\tan(11^\circ)=0.1944\). The difference between \(10^\circ\) and \(11^\circ\) is \(1^\circ = 60'\), and we have \(35'\) which is \(35/60\) of the way from \(10^\circ\) to \(11^\circ\). The difference in tangent values is \(0.1944 - 0.1763 = 0.0181\). So the additional tangent for \(35'\) is \(0.0181\times(35/60)\approx0.0181\times0.5833\approx0.01056\). So \(\tan(10^\circ35')\approx0.1763 + 0.01056\approx0.18686\). Then \(h = 324\times0.18686\approx324\times0.18686\). Let's calculate that: \(324\times0.1 = 32.4\), \(324\times0.08 = 25.92\), \(324\times0.00686 = 324\times0.006 + 324\times0.00086 = 1.944+0.27864 = 2.22264\). So adding up: \(32.4 + 25.92 = 58.32 + 2.22264 = 60.54264\approx61\)? Wait, but maybe using a calculator directly: \(\tan(10^\circ35')=\tan(10 + 35/60)^\circ=\tan(10.583333\cdots^\circ)\approx0.1869\). Then \(324\times0.1869 = 324\times0.18 + 324\times0.0069 = 58.32 + 2.2356 = 60.5556\approx61\). Wait, but the options have 61 as option B. Wait, maybe I made a mistake in the angle conversion? Wait, no, let's check with a calculator. Let's use a calculator to compute \(\tan(10^\circ35')\). Let's convert \(35'\) to degrees: \(35\div60\approx0.5833\), so \(10.5833^\circ\). Using a calculator, \(\tan(10.5833^\circ)\approx0.1869\). Then \(324\times0.1869 = 324\times0.1869\). Let's do this multiplication: \(324\times0.1869\). Let's compute \(300\times0.1869 = 56.07\), \(24\times0.1869 = 4.4856\), so total is \(56.07 + 4.4856 = 60.5556\approx61\). So the height is approximately 61 feet.

Answer:

B. 61 ft