Q 127. - Precalculus Enhanced with Graphing Utilities

Question

Calculating the Time of a Trip Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a 

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

See the figure.

Sally can jog 8 miles per hour along the paved road, but only 3 miles per hour in the sand on the beach. Because of a river directly between the two houses, it is necessary to jog in the sand to the road, continue on the road, and then jog directly back in the sand to get from one house to the other. The time T to get from one house to the other as a function of the angle $\theta$  shown in the illustration.

$T\left(\theta \right)=1+\frac{2}{3 \sin \theta }-\frac{1}{4 \tan \theta }$and $\theta$ lies between $0$ to $90°$.

(a)  

Calculate the time T for $\theta =30°$. How long is Sally on the paved road?

(b)  

Calculate the time T for $\theta = 45°$. How long is Sally on the paved road?

(c) 

Calculate the time T for $\theta =60°$ How long is Sally on the paved road?

(d) 

Calculate the time T for $\theta =90°$. Describe the path taken. Why can't the formula for T be used?

Step-by-Step Solution

Answer

(a) T for $\theta =30°$is $\frac{28-3\sqrt{3}}{12}$.

(b) T for $\theta =45°$ is $\frac{15+8\sqrt{2}}{12}$ .

(c) T for $\theta =60°$ is $\frac{36+\sqrt{3}}{36}$.

(d) T for $\theta =90°$ is $\frac{5}{3}$

Step 1. Finding value for θ = 30 ° .

Substitute the value of $\theta$ in the given formula.

$T\left(\theta \right)=1+\frac{2}{3 \sin \theta }-\frac{1}{4 \tan \theta }$

$$\begin{gathered} T\left(30°\right)=1+\frac{2}{3 \sin 30°}-\frac{1}{4 \tan 30°} \\ =\frac{28-3\sqrt{3}}{12} \end{gathered}$$

Step 2. Finding the value for θ = 45 ° .

Substitute the value of $\theta$ in the given formula.

$$\begin{gathered} T\left(\theta \right)=1+\frac{2}{3 \sin \theta }-\frac{1}{4 \tan \theta } \\ T\left(45°\right)=1+\frac{2}{3 \sin 45°}-\frac{1}{4 \tan 45°} \\ =\frac{15+8\sqrt{2}}{12} \end{gathered}$$

Step 3. Finding the value for θ = 60 ° .

Substitute the value of $\theta =60°$in the given formula .

$$\begin{gathered} T\left(\theta \right)=1+\frac{2}{3 \sin \theta }-\frac{1}{4 \tan \theta } \\ T\left(60°\right)=1+\frac{2}{3 \sin 60°}-\frac{1}{4 \tan 60°} \\ =\frac{36+\sqrt{3}}{36} \end{gathered}$$

Step 4. Finding the value for θ = 90 ° .

Substitute the value of$\theta =90°$ in the given formula.

$$\begin{gathered} T\left(\theta \right)=1+\frac{2}{3 \sin \theta }-\frac{1}{4 \tan \theta } \\ T\left(90°\right)=1+\frac{2}{3 \sin 90°}-\frac{1}{4 \tan 90°} \\ =\frac{5}{3} \end{gathered}$$