A family is building a circular fountain in the backyard the yard is rectangular and measures 6x by 7x and the fountain is going to be circular with a radius of 2x once the fountain is built what will be the area of the remaining yard?

Ac + Ar = Ay, pi*r^2 + Ar = 6x*7x, 3.14*(2x)^2 + Ar = 42x^2,

12.57x^2 + Ar = 42x^2,
Ar = 29.4x^2 = Area remaining.

the answer on your TEST by the way i'm not supposed to help you with but i'll let it slide it is Option D.

To find the area of the remaining yard after building the circular fountain, we need to first calculate the area of the entire yard and then subtract the area of the circular fountain.

The first step is to find the area of the rectangular yard. The yard's dimensions are given as 6x by 7x, which means the length of the yard is 6x and the width is 7x. To find the area, we multiply the length by the width:

Area of the rectangular yard = length x width = 6x * 7x

Now, let's move on to calculate the area of the circular fountain. The radius of the circular fountain is given as 2x. The formula to find the area of a circle is:

Area of a circle = π * radius^2

Substituting the value of the radius into the formula, we get:

Area of the circular fountain = π * (2x)^2

Simplifying further:

Area of the circular fountain = π * 4x^2

Now, to find the area of the remaining yard, we subtract the area of the circular fountain from the area of the rectangular yard:

Area of remaining yard = Area of rectangular yard - Area of circular fountain
= (6x * 7x) - (π * 4x^2)

Simplifying further, we have:

Area of remaining yard = 42x^2 - 4πx^2

Therefore, the area of the remaining yard, after building the circular fountain, is given by 42x^2 - 4πx^2.