# Thread: Maximum area of an area within y=x^2

1. ## Maximum area of an area within y=x^2

I've got this problem that I'm trying to solve, I've been twisting my brain for several hours now.

An area 'A' is limited by the curve y=x^2, the x-axel and the line x=2

In this area is a rectangle with the area R, as seen in the figure.

Calculate the maximum relation between R and A in exact form.

If someone could take a look at this and help me, I would appreciate it very very much!

2. if you make the left hand bottom corner of the rectangle point a then it follows that

$\displaystyle A = L\times W$

subing in some values gives

$\displaystyle A = f(a)\times (2-a)$

$\displaystyle A = a^2\times (2-a)$

$\displaystyle A = 2a^2-a^3$

for a maximum you must find where $\displaystyle \frac{dA}{da}=0$

$\displaystyle \frac{dA}{da}=4a-3a^2$

$\displaystyle 0=4a-3a^2$

$\displaystyle 0=a(4-3a)$

$\displaystyle a=0,\frac{4}{3}$

probably should discard zero as a solution at this point as it implies zero area.

now $\displaystyle f\left(\frac{4}{3}\right)= \frac{9}{16}$

Maximum area should be simple to find from here using

$\displaystyle A = L\times W$

$\displaystyle A = f(a)\times (2-a)$