A family is having a pool built in their backyard. If their yard in rectangular and measures 10x by 5x and the pool is circular with a radius of 2x how much of the yard will be left over after the pool is built? Write your answer in factored form.
I assume you know the formulas for area of rectangle and circle.
Now just subtract the circle from the rectangle. Since both expressions are for area, there should be some x^2 in each term that you can factor out.
To find out how much of the yard will be left over after the pool is built, we need to calculate the area of the pool and subtract it from the total area of the yard.
The area of a rectangular yard is given by multiplying its length by its width. In this case, the length of the yard is 10x and the width is 5x, so the total area of the yard is (10x) * (5x) = 50x^2.
The area of a circular pool is given by the formula A = πr^2, where A is the area and r is the radius. In this case, the radius of the pool is 2x, so the area of the pool is π(2x)^2 = 4πx^2.
Now, we can calculate the area of the yard that will be left over by subtracting the area of the pool from the total area of the yard:
Leftover area = Total area of yard - Area of pool
= 50x^2 - 4πx^2
Since we are asked to write the answer in factored form, we can factor out the common term x^2:
Leftover area = x^2(50 - 4π)
Therefore, the amount of yard left over after the pool is built can be represented in factored form as x^2(50 - 4π).