Calculus Optimization Problem

drasord

New member
I'm really stuck on this problem. Could anyone provide some help?

Find the length and width of the rectangle of largest area that can be inscribed in a semicircle of radius R, assuming that one side of the rectangle lies on the diameter of the semicircle. Also, find the area of this rectangle. Draw a neat diagram.

Thanks!

MarkFL

Staff member
Can you show us what you have tried so far so we know where you are stuck and can best offer help?

MarkFL

Staff member
You have found the correct critical value:

$$\displaystyle x=\frac{R}{\sqrt{2}}$$

The base of the rectangle is $2x$. The height is $y$.

So, what is $$\displaystyle y\left(\frac{R}{\sqrt{2}} \right)$$ ?

Last edited:

drasord

New member
Or now I have to find area:

A = l * w
A = 2R/sqrt(2) * sqrt(R^2/2)
A = sqrt(2) * sqrt(x) * sqrt(x^2)

?

MarkFL

$$\displaystyle 2x=2\cdot\frac{R}{\sqrt{2}}=\sqrt{2}R$$
$$\displaystyle y=\sqrt{R^2-\frac{R^2}{2}}=\frac{R}{\sqrt{2}}$$
$$\displaystyle A=\sqrt{2}R\cdot\frac{R}{\sqrt{2}}=R^2$$