Chapter 7.5, Problem 34E

a.

To determine

To write: and graph a system of inequalities that describes the requirements of the facility.

Answer to Problem 34E

The system of inequality is:

  $\begin{array}{l}xy\ge 500\\ 2x+\pi y\ge 125\\ x\ge 0\\ y\ge 0\end{array}$

The graph will be

  

Explanation of Solution

Given

A physical fitness facility is constructing an indoor running track with space for exercise equipment inside the track. The track must be at least 125 meters long, and the exercise space must have an area of at least 500 square meters.

  

Concept used:

Area of rectangle: length $\times$ breadth

Radius of the circle is $\frac{\text{Diameter}}{2}$

Calculation:

Let x be the length and y be the breadth

The area inside the rectangle must be at least 500 square meters, so

  $xy\ge 500$

The length of track must be at least 125m

The length of the track consists of two semicircles at the breadth side and a rectangle in middle.

Here breadth is equal to diameter of the semicircles

Since we have two semicircles having equal radius

The length of the track will be $2\times \frac{y}{2}\times \pi +2\times x=\pi y+2x$

From the given condition $2x+\pi y\ge 125$

The dimensions (length and breadth) are always greater or equal to 0

So, $x\ge 0,y\ge 0$

The system of inequality is:

  $\begin{array}{l}xy\ge 500\\ 2x+\pi y\ge 125\\ x\ge 0\\ y\ge 0\end{array}$

Here $x\ge 0,y\ge 0$ means the region above x-axis and the region on the right of y-axis

  $xy\ge 500$ will be the region where product of $xy$ is greater than or equal to 500.

  $2x+\pi y\ge 125$ will be the region above the line $2x+\pi y=125$ because point (0,0) is not included in the region.

The table for $2x+\pi y=125$ will be

  $\begin{array}{cccc}x& 0& \frac{125}{2}& 62\\ y& \frac{125}{\pi }& 0& \frac{1}{\pi }\end{array}$

Plot the points and join them along with all the conditions mentioned above

  

b.

To determine

To calculate: two solutions of the system and interpret their meanings in the context of the problem.

Answer to Problem 34E

The two solutions are $\left(40,40\right)$ and $\left(40,30\right)$

Explanation of Solution

Given information

A physical fitness facility is constructing an indoor running track with space for exercise equipment inside the track. The track must be at least 125 meters long, and the exercise space must have an area of at least 500 square meters.

The systems of equations are

  $\begin{array}{l}xy\ge 500\\ 2x+\pi y\ge 125\\ x\ge 0\\ y\ge 0\end{array}$

The solution to the above system is

  

Concept used:

Any points on the solution region can be solutions of the system

Calculation:

From the graph we can see that $\left(40,40\right)$ and $\left(40,30\right)$ lies in the region.

Hence $\left(40,40\right)$ and $\left(40,30\right)$ are the two solutions of the system

For $\left(40,40\right)$ the length is 40m and the breadth is 40m

The area of the space will be $40\times 40=1600{\text{m}}^2$ which is more than the minimum requirement

For $\left(40,30\right)$ the length is 40m and the breadth is 30m

The area of the space will be $40\times 30=1200{\text{m}}^2$ which is more than the minimum requirement