Math

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.

  1. 👍
  2. 👎
  3. 👁
  1. i dont have an answer but just pointing out that 256/16 is 16 not 15 so xandy are both 16

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. maths

    the length of a rectangle is 4 times than its width. the area of the rectangle is 1024cm2 find the length of the rectangle the perimeter of the rectangle 4 square has the same perieter as that its rectangle. Find the area of the

  2. Calculus

    A rectangle is bounded by the x-axis and the semicircle y=ã(25-x^2). Question is, what length and width should the rectangle have so that its area is a maximum, and what is the maxuimum area? Area= length*width = 2x*y=

  3. Calculus

    A rectangle has area 64 m2. Express the perimeter of the rectangle as a function of the length L of one of its sides. State the domain of P. (Assume the length of the rectangle is larger than its width. Enter your answer using

  4. Algebra I

    The expression 2(l+w) may be used to find the perimeter of a rectangle. What are the length and width of a rectangle if the area is 13 1/2 square units and the length of one side is 1/5 the measure of the perimeter?

  1. Math

    The sides of the rectangle are in the ratio of 4:7. If its length is 31.5 in., find the width, the perimeter, and the area of this rectangle.

  2. Mathematics

    The length of a rectangle is four times its width. If the perimeter of the rectangle is 80yd, find its area.

  3. algebra

    The length of a rectangle is 4 inches more than its width. The area of the rectangle is equal to 5 inches more than 2 times the perimeter. Find the length and width of the rectangle.

  4. Math

    If the length of a rectangle is (2x-1)cm and its width is (x+3)cm. How do i write an expression in the form ax^2+bx+c for the area of the rectangle? Given that the area of the rectangle is 294cm^2, determine the value of x and

  1. Math 2

    The rectangle below has an area of 30k^3+6k^2 The width of the rectangle (in meters) is equal to the greatest common monomial factor of 30k^3 and 6k^2 What is the width and length of the rectangle? Width:6k^2 Length: I couldn't

  2. Math

    The length of a rectangle is twice its width if the area of the rectangle is 200 m squared, find its perimeter.

  3. Algebra

    The length of a rectangle is six times its width. If the perimeter of the rectangle is 140ft, find its area.

  4. Math

    A norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window with maximum area if the total perimeter is 16 feet. X = the width of the rectangle. Y =

You can view more similar questions or ask a new question.