Chapter 8.2, Problem 6STP

A rectangular blacktop with a length of 5x and a width of 3x has been erected inside a rectangular field that has a length of 12x and a width of 7x

What is the area of the part of the field that is not blacktop?

b. There is a circular fountain in the rectangular field that has a radius of 3x. What is the area of the part of the field that does not include the blacktop or the fountain? Factor your answer.

To determine

To determine the area of the part of the field that is not blacktop.

Answer to Problem 6STP

Area of field that is not blacktop is $69x^2$

Explanation of Solution

Given information :

Diamentions of rectangular blacktop is $5x\times 3x$

Diamentions of rectangluar field is $12x\times 7x$

Area of the part of the field that is not blacktop = area of field − area of blacktop.

Now we shall put the values we get,

Area of field that is not blacktop $=\left(12x\times 7x\right)-\left(5x\times 3x\right)$

Hence we get Area of field that is not blacktop is $84x^2-15x^2=69x^2$

To determine

To determine the area of the part of the field that does not include the blacktop or the fountain.

Answer to Problem 6STP

Required area is $3x^2\left(23-3\pi \right)$

Explanation of Solution

Given information :

Radius of circular fountain is $3x.$

Area of the part of the field that without blacktop and fountain is,

Area of field − area of blacktop − area of circular fountain.

Hence we get,

  $\begin{array}{l}A=\left(12x\times 7x\right)-\left(5x\times 3x\right)-\left(\pi {\left(3x\right)}^2\right)\\ A=84x^2-15x^2-\pi 9x^2\\ A=69x^2-\pi 9x^2\\ A=3x^2\left(23-3\pi \right)\end{array}$

Hence required area is $3x^2\left(23-3\pi \right)$