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?

drasord

New member
Absolutely - sorry for the delay! This is my understanding of how to "optimize" the problem:

So I've found the area, I believe. But I need to find the "rectangle of largest area that can be inscribed in a semicircle of radius R". I'm confused about how to do this. And what does the professor mean by "a neat diagram"?

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 $$\displaystyle y\left(\frac{R}{\sqrt{2}} \right)$$ ?

drasord

New member

So now I have A, length, and width. Correct?

A = $$\displaystyle 2x * sqrt(R^2 - x^2)$$

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

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