Max/Min Problem: Find Max Area of Rectangle Bounded by X-Axis & Semicircle

[Riogho]

Homework Statement

A rectangle is bounded by the X-axis and the semicircle Y = [(sqrt)36-x^2]. What dimensions should the rectangle have so that its area is a maximum.

Homework Equations

Just a note, the 36-x^2 is all under the radical.

The Attempt at a Solution

Ditto.

 

[gabbagabbahey]

Well, the width of the rectangle is clearly $2x$, and its height is $y=\sqrt{36-x^2}$...so what is its area $A(x)=?$ for a given value of $x$? How do you find the maximum of such a function?

P.S. I don't think the definition of "ditto" is exactly what you seem to think it is (just a friendly FYI)

 

[Riogho]

Wouldn't I just find the area by multiplying both of them together?

 

[gabbagabbahey]

Sounds good to me; the last time I checked the area of a rectangle was just width times height ;0)

 

[Riogho]

So I multiply them together then take the derivative, and then find the critical points of that?

 

[HallsofIvy]

Yes, now do it!

 

[Riogho]

Done.

Many thanks.