A rectanglar 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? (I believe the answer is 69x^2)

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 of the fountain? Factor your answer. (Please show me how to set up and solve) Thanks

5x3 = 15

12x7 = 84
84-15 is 69, so you are corect

area of circle is pi(3x)^2 = 9pi x^2
so, subtract: 84x^2 - 9pi x^2 = (84-9pi)x^2

the factor of part b. would end up being 3x^2(23-3pi). because you would have to bring down the area of part a. which would be 69x^2 with the 9pix^2 not 84, therefore, giving the factored solution of 3x^2(23-3pi). 3 is the gcf because 69 cannot and 9 cannot be multiplyed to obtain each with the same nuber other than 3.

a)84-15=69

Area=69x
b) 3x^2(84-pi)

To find the area of the part of the field that is not covered by the blacktop, you need to find the area of the field and subtract the area of the blacktop.

The area of a rectangle can be calculated by multiplying its length and width. The area of the rectangular field is given as:

Area of field = length × width = (12x) × (7x) = 84x^2

Similarly, the area of the blacktop is given by the product of its length and width:

Area of blacktop = length × width = (5x) × (3x) = 15x^2

Now, to find the area of the part of the field that is not blacktop, we subtract the area of the blacktop from the area of the field:

Area not covered by blacktop = Area of field - Area of blacktop
= 84x^2 - 15x^2
= 69x^2

So, the area of the part of the field that is not blacktop is indeed 69x^2.

Now, let's calculate the area of the part of the field that does not include the blacktop of the fountain.

The area of a circle can be calculated using the formula: Area = π × radius^2.

Given that the radius of the fountain is 3x, we can calculate its area:

Area of fountain = π × (3x)^2
= π × 9x^2
= 9πx^2

To find the area of the field excluding the blacktop of the fountain, we need to subtract the area of the fountain from the area of the field:

Area not covered by blacktop and fountain = Area of field - Area of fountain
= 84x^2 - 9πx^2

The expression 9πx^2 cannot be factored further as it is the product of a constant (9π) and a variable (x^2).

So, the area of the part of the field that does not include the blacktop of the fountain is 84x^2 - 9πx^2.