Posted by **l** on Thursday, August 31, 2006 at 10:18pm.

Find the Maximum area for the given perimeter of a rectangle. State the length and width of the rectangle.

28 inches

Well, finally a calculus problem.

Ok, we know that the area for a rectangle is

A=l*w and the perimeter is P=2(l+w)

In this problem P= 28, so let's express one of the dimensions in terms of the other an substitute into the area formula. Thus 28=2(l+w). Let's solve for l in terms of w, thus 14=l+w, or l=14-w When we substitute this into the area formula we get

A=(14-w)*w. So A=14w-w^2

Now find dA/dw and evaluate the critical points.

dA/dw = 14-2w and dA/dw = 0 means 14-2w=0

So w = 7 is a critical point. I'll let you verify that this is the max. (the second deriv is -2, what does that mean?)

Thus the rectangle of maximum area has w=7. If you put this back into the formula for the perimeter, you'll find that l=7 too. This means that the rectangle of max. area is a square with a side=P/4.

Considering all rectangles with the same perimeter, the square encloses the greatest area.

Proof: Consider a square of dimensions x by x, the area of which is x^2. Adjusting the dimensions by adding a to one side and subtracting a from the other side results in an area of (x + a)(x - a) = x^2 - a^2. Thus, however small the dimension "a" is, the area of the modified rectangle is always less than the square of area x^2.

This should give you your answer.

## Answer this Question

## Related Questions

- area, perimeter - the perimeter of rectangle y is equal to its area. rectangle z...
- math - the perimeter of rectangle y is equal to its area. rectangle z has the ...
- Math - The first question is this: Helen designs a rectangle with an area of 225...
- Math - Determine the perimeter and area of a rectangle that has a width of 6 ...
- Math - The area of a rectangle is shown below The area of the rectangle is 56 ...
- basic geometric - 10. a rectangle has width the same as a side of a square whose...
- Math - The perimeter of rectangle Z is equal to its area.Rectangle Y has the ...
- writing to explain - The perimeter of rectangle Y is equal to its area. ...
- math117 - The area of a rectangle of length x is given by 3x2 +5x. Find the ...
- math - a rectangle is twice as long as it is wide. if both of its dimensions are...

More Related Questions