A family is building a circular fountain in the backyard. The yard is rectangular and measures 7x by 6x 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?
Can someone please walk me through this to help me understand? Thanks :)
yard area = 42 x^2
fountain area = pi r^2 = pi (4x^2)
so remaining area = (42 - 4pi)x^2
To find the area of the remaining yard after the circular fountain is built, we need to first calculate the area of the entire rectangular yard and then subtract the area of the circular fountain.
Let's start by finding the area of the rectangular yard. The formula for the area of a rectangle is length times width, so in this case, it would be:
Area of the rectangular yard = length * width
Given that the length is 7x and the width is 6x, we can substitute these values into the formula:
Area of the rectangular yard = 7x * 6x
Now, let's simplify this expression:
Area of the rectangular yard = 42x^2
Next, we need to find the area of the circular fountain. The formula for the area of a circle is pi times the radius squared. In this case, the radius is 2x, so the area of the circular fountain would be:
Area of the circular fountain = pi * (radius)^2
Substituting the value of the radius:
Area of the circular fountain = pi * (2x)^2
Simplifying this expression:
Area of the circular fountain = 4pi x^2
Finally, 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 the remaining yard = Area of the rectangular yard - Area of the circular fountain
Substituting the previously calculated values:
Area of the remaining yard = 42x^2 - 4pi x^2
This is the formula for finding the area of the remaining yard after the circular fountain is built. You can simplify this expression further if desired.
Hm well I know the right answer but I need to know how to get there. It says the answer is 2x^2(21-2pi)
see
[(42(x^2)-4pi(x^2)]
21*2(x^2)=42(x^2) rit?
2*2(x^2)pi=4(x^2)pi rit?
[(21*(2x^2)-(2x^2)*2pi)]
bring (2x^2) out
you get
(2x^2)(21-2pi)
do you undestand now