Q 119 - Precalculus Enhanced with Graphing Utilities

Question

Calculating the Time of a Trip  From a parking lot, you want to walk to a house on the beach. The house is located $1500$ feet down a paved path that parallels the ocean, which is $500$ feet away. See the illustration. Along the path you can walk $300$ feet per minute, but in the sand on the beach you can only walk 100 feet per minute.

        The time $T$ to get from the parking lot to the beach house can be expressed as a function of the angle $\theta$ shown in the illustration and is

$T\left(\theta \right)=5-\frac{5}{3\tan \theta }+\frac{5}{\sin \theta }, 0<\theta <\frac{\mathrm{\pi }}{2}$

Calculate the time $T$ if you walk directly from the parking lot to the house.

Step-by-Step Solution

Answer

The time of $15.82approx.$ minutes will taken if we walked directly from the parking lot to the house.

Step 1. Given Information

We have given that a house is located $1500$ feet down a paved path that parallel the ocean which is $500$ feet away. Along the path we can walk $300ft$ per minute, but in sand $100ft$ per minute.

The time $T$ to get from parking lot to house as a function $\theta$ is shown as following function :-

$T\left(\theta \right)=5-\frac{5}{3\tan \theta }+\frac{5}{\sin \theta }, 0<\theta <\frac{\mathrm{\pi }}{2}$

We have to calculate the time $T$ if we walk directly from parking lot to house.

Step 2. Calculating required time

We have given the following function of $\theta$ for time to get from parking lot to the beach house as following :-

$T\left(\theta \right)=5-\frac{5}{3\tan \theta }+\frac{5}{\sin \theta }, 0<\theta <\frac{\mathrm{\pi }}{2}$

If we walk directly walk from the parking lot to the house, then by using trigonometry ratio for a right angled triangle, we have :-

$$\begin{gathered} \tan \theta =\frac{500}{1500} \\ \Rightarrow \tan \theta =\frac{1}{3} \\ \Rightarrow \theta ={\tan }^{-1}\left(\frac{1}{3}\right) \\ \Rightarrow \theta =18.4 \end{gathered}$$

Put this value of $\theta$ in the given time function, then we have :-

$$\begin{gathered} T\left(\theta \right)=5-\frac{5}{3\tan \left(18.4\right)}+\frac{5}{\sin \left(18.4\right)} \\ \Rightarrow T\left(\theta \right)=5-\frac{5}{3\times {\displaystyle \frac{1}{3}}}+\frac{5}{0.316} \\ \Rightarrow T\left(\theta \right)=5-\frac{5}{1}+15.82 \\ \Rightarrow T\left(\theta \right)\approx 5-5+15.82 \\ \Rightarrow T\left(\theta \right)\approx 15.82 \end{gathered}$$

So it will take approximately$15.82$ minutes if we walked directly from the parking lot to the house.