Chapter 1.6, Problem 84E

Distance, Speed, and Time A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A man is standing on the boardwalk, exactly 750 ft across the sand from his beach umbrella, which is right at the shoreline. The man walks 4 ft/s on the boardwalk and 2 ft/s on the sand. How far should he walk on the boardwalk before veering off onto the sand if he wishes to reach his umbrella in exactly 4 min 45 s?

To determine

To find: The distance covered on the boardwalk.

Answer to Problem 84E

The distance covered on the boardwalk is 720 ft.

Explanation of Solution

Given:

Distance between boardwalk and shoreline is 210 ft. Distance between man and umbrella is 750 ft.

The man walks 4 ft/s on boardwalk and 2 ft/s on the sand.

Time taken to reach the umbrella is exactly 4 min 45 sec.

Calculation:

Let the distance covered on the boardwalk be x.

Time taken to reach the umbrella in sec is,

$\begin{array}{c}4\min 45\sec =\left(4\times 60+45\right)\sec \\ =\left(240+45\right)\sec \\ =285\sec \end{array}$

Tabulate the given information algebraically.

In words	In algebra
Length of boardwalk	x

Model the equation for the above information.

$\begin{array}{c}{\left(\begin{array}{l}\text{Distance}\text{of}\text{man}\\ \text{from}\text{umbrella}\end{array}\right)}^2={\left(\begin{array}{l}\text{Distance}\text{cover}\\ \text{on}\text{boardwalk}\end{array}\right)}^2+{\left(\begin{array}{l}\text{Distance}\text{of}\text{boardwalk}\\ \text{from}\text{umbrella}\end{array}\right)}^2\\ {\left(750\right)}^2={\left(x\right)}^2+{\left(210\right)}^2\end{array}$

On further simplification,

$\begin{array}{l}562500=x^2+44100\\ x^2=518400\\ x=720\end{array}$

The distance covered on board walk is 720 ft.

Time taken to walk through the board-walk to a point directly in front umbrella is,

$\begin{array}{c}\text{total}\text{time}=\text{time}\text{on}\text{boardwalk}+\text{time}\text{on}\text{sand}\\ =\frac{720}{4}+\frac{210}{2}\\ =180+105\\ =285\sec \end{array}$

Thus, the distance cover on the boardwalk is 720 ft.