Maximum area and minimum perimeter.

R

Rumor

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Sep 8, 2009
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I'm stuck on this problem.

"A rectangle has one side on the x-axis, one side on the y-axis, one vertex at the origin and one on the curve y=e^-2x (greater than or equal to) 0. Find the

a) Maximum area
b) Minimum perimeter."

Here's a picture of the curve:

example.jpg


I'm not sure how to start this one off. Any help on how to do this would be appreciated!
 
The black point's horizontal position is x, which gives the width of the rectangle.
It's height is f(x), that's it's distance above the x-axis.
The rectangle area is (height)(width).

If you draw the curve of this area, it is the graph of (x)[f(x)].

How do you locate the maximum value of such a curve?

Also, it's perimeter is x+f(x)+x+f(x).

Perform the same procedure for the maximum value of this.

At the maximum value, the slope of the tangent to the area and perimeter graphs will be zero.
How do you calculate the slope of the tangent using calculus?
 
Thank you fasteddie, as I have been guilty of this social blunder in the past. Will now hopefully rectify the situation.
 
Sorry about the typo regarding "minimum perimeter" !
I meant minimum perimeter of course.

There is no need to find the 2nd derivatives of the area and perimeter equations,
to distinguish between maximums and minimums in a case like this.

The area has a single maximum, in between the two extremes of zero.
The perimeter has the potential to be half the x-axis, graph has a single minimum.
 
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