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. 👍 0
  2. 👎 0
  3. 👁 197
asked by l
  1. i dont have an answer but just pointing out that 256/16 is 16 not 15 so xandy are both 16

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. basic geometric

    10. a rectangle has width the same as a side of a square whose perimeter is 20m. the length of the rectangle is 9m. find the perimeter of this rectangle. 34. The width of a rectangular picture is one-half the length. The perimeter

    asked by vero on March 12, 2007
  2. basic geometry

    A rectangle has width the same asa a side of a square whose perimeter is 20m.the length of the rectangle is 9m. Find the perimeter of this rectangle. 34. The with of a rectangular picture is on-half the length . The perimeter of

    asked by vero on March 13, 2007
  3. Math

    The area of a rectangle is shown below The area of the rectangle is 56 Which of the following are possible values for the length and width of the rectangle? A.length=6cm and width=9cm B.length=7cm and width=8cm C.length=14cm and

    asked by Jerald on March 6, 2013
  4. math,help

    help Probblem #6 solve by multiplying through with the common denominator (11)/(3)x + (1)/(6) = (8)/(3)x + (19)/(6) Problem #20 the perimeter of a rectangle is to be no greater than 300in. and the length must be 125in. write an

    asked by jasort20 on March 18, 2007
  5. maths

    the length of a rectangle is 3 less than 5 times its width. Write a simpified algebriac expression for the perimeter of a rectangle. If the rectangle width is tripled and its length is doubled,the perimeter of new rectangle is

    asked by aisha on November 24, 2010
  6. Math

    The first question is this: Helen designs a rectangle with an area of 225 square units. Her rectangle is the largest rectangle (that is, with largest area) with whole-number side lengths that can be made from the perimeter of the

    asked by Gabby on February 25, 2013
  7. 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 =

    asked by Tezuka on September 17, 2006
  8. algebra

    1. What must be added to p to produce q? 2. What is the perimeter of the rectangle if the length is 2x-3y and the width is x+2y? What is the area? 3. What must be subtracted to 7y to leave -5y+2z? 1. q = 1 - p 2. Perimeter of

    asked by klar maesen on July 18, 2007
  9. 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

    asked by Breauna on August 25, 2014
  10. geometric

    The perimeter of a rectangular carpet is 70feet. The with is three -fourths the length .Find the width. 38. The length of a rectagular room is six times as the width . The perimeter 84 yards. find the width. 39. The width of a

    asked by baby girl on March 13, 2007

More Similar Questions