Q. 120 - Precalculus Enhanced with Graphing Utilities

Question

Two oceanfront homes are

located 8 miles apart on a straight stretch of beach, each a

distance of 1 mile from a paved path that parallels the ocean.

Sally can jog 8 miles per hour on the paved path, but only 3

miles per hour in the sand on the beach. Because a river flows

directly between the two houses, it is necessary to jog in the

sand to the road, continue on the path, and then jog directly

back in the sand to get from one house to the other. See the

illustration. The time T to get from one house to the other as

a function of the angle $\theta$ shown in the illustration is

$T\left(\theta \right) = 1+\frac{2}{3Sin\theta }-\frac{1}{4\tan \theta } 0<\theta <\frac{\mathrm{\pi }}{2}$

(a) Calculate the time T for tan$\theta =\frac{1}{4}$

(b) Describe the path taken.

(c) Explain why $\theta$ must be larger than 14°.

Step-by-Step Solution

Answer

(a) Time taken will be 2.75 hours.

(b) Sally has to jog two times in sand and cross a bridge.

(c) The angle must be greater than ${14}^{\circ }$ inorder for her to cross the bridge.

Part (a) Step 1. Given Information

$T\left(\theta \right) = 1+\frac{2}{3Sin\theta }-\frac{1}{4\tan \theta } 0<\theta <\frac{\mathrm{\pi }}{2}$

Part (a) Step 2. Calculation

$$\begin{gathered} From the question \\ \tan \theta =\frac{1}{4} \\ Now we will calculate Sin\theta \\ from the figure \\ Sin\theta =\frac{y}{r} \\ where, \\ r=\sqrt{x^2+y^2} \\ therefore, \\ \sin \theta =\frac{y}{\sqrt{x^2+y^2}} \\ and \sqrt{x^2+y^2}=\sqrt{4^2+1^2} \\ =\sqrt{17} \\ \sin \theta =\frac{1}{\sqrt{17}} \\ substituting this in the equation for time we get \\ T\left(\theta \right)=1+\frac{2}{3\sin \theta }-\frac{1}{4\tan \theta }, 0<\theta <\frac{\mathrm{\pi }}{2} \\ T\left(\theta \right)=1+\frac{2}{3\times {\displaystyle \frac{1}{\sqrt{17}}}}-\frac{1}{{\displaystyle \frac{1}{4}}} \\ =1+\frac{2\sqrt{17}}{3}-1 \\ =2.75 hours \end{gathered}$$

Part (b) Step 1. Calculation

Sally chooses a path where she needs to jog in the sand then cross the bridge and again jog in the sand to reach her destination.

Part (c) Step 1. Explanation

The angle must be greater than ${14}^{\circ }$ inorder for her to cross the bridge.