a circular window with a circumference of 10 feet is to be placed into a wall at a new home. The builder cuts a hole for the window and says that it has an area of at least 8 square feet. explain why she is right or wrong. Show work.

Builder is right.

Step-by-step explanation:

circumference, C = 10 feet

Let the radius of circular window is r.

Use the formula for circumference.

C = 2 x π x r = 10 ... (1)

10 = 2 x π x r

r = 1.59 ft

The formula for the area of circular window

A = π r²

Differentiate with respect to r.

dA/dr = 2 x π x r

Differentiate again with respect to r.

d²A/dr² = 2 π

It is positive for any value of r so the area of minimum.

So, minimum area is

A = 3.14 x 1.59 x 1.59 = 8 ft

So, the minimum area id 8 ft². The builder is right.