Chapter 8.3, Problem 67PS

Solution

To calculate : The distance at which sprinklers should be placed to water the entire flower bed with least possible overlap in coverage.

The sprinklers should be placed $7.94$ feet apart.

Given information :

A row of sprinklers is to be installed parallel to and $4.5$ feet away from the back edge of a flower bed. Each sprinkler waters a region with a $6$ foot radius.

  

Formula used:

Pythagoras Theorem:

  ${\left(\text{Hypotenuse}\right)}^2={\left(\text{Perpendicular}\right)}^2+{\left(\text{Base}\right)}^2$

Explanation :

Redraw the given figure as below:

  

Consider right angle in the given figure. Consider the right triangle in the figure. Use the Pythagorean Theorem.

Thus,

  $\begin{array}{c}{6}^2={\left(4.5\right)}^2+x^2\\ 36=20.25+x^2\\ 36-20.25=x^2\\ 15.75=x^2\end{array}$

Take square root of both the sides.

  $\begin{array}{c}\sqrt{15.75}=\sqrt{x^2}\\ 3.97=x\end{array}$

The distance between the sprinklers is $2x$ .

  $\begin{array}{c}2x=2\left(3.97\right)\\ =7.94\end{array}$

Therefore, the sprinklers should be placed $7.94$ feet apart.