Maximizing area of rectangle inscribed in semi circle

[Guest]

Find the area of the largest rectangle that can be inscribed inside a semi circle with a radius of 10 units. Place the length of the rectangle along the diameter.

How do you figure this out? I'm tihnking you have to use trig, but what's the formula for it? I forget.

Thanks for the help + time

 

[TchrWill]

Find the area of the largest rectangle that can be inscribed inside a semi circle with a radius of 10 units. Place the length of the rectangle along the diameter.

How do you figure this out? I'm tihnking you have to use trig, but what's the formula for it? I forget.

Thanks for the help + time

The rectangular figure of greatest area within a circle is a square.

Therefore, the rectangular figure of greatest area within a semi-circle is one half of that square.

The maximum full square has area A = [2(10)(sqrt2)/2]^2 =

The rectangle of maximum area within the semi-circle is therefore, [2(10)(sqrt2)/2]^2/2 = [20(sqrt2)/2]^2/2.

 

[skeeter]

first, draw a sketch

base of the rectangle that lies on the x-axis is \(\displaystyle 2x\)

height of the rectangle is \(\displaystyle y = \sqrt{10^2 - x^2}\) {equation for a semicircle of radius 10, remember?}

\(\displaystyle A = 2x\sqrt{100 - x^2}\)

find \(\displaystyle \frac{dA}{dx}\) and maximize.