A rectangular swimming pool of length l and width w has a perimeter of 64 feet.

Question

a) Write an expression that represents that area of the swimming pool in terms of w.

b) Use your expression from part(a) to determine the dimensions of the pool that maximize the area. What do you notice?

 

I'm really confuse, don't know where to start. Please help.

Answer 1

 

 P = 2L + 2w

 

64 = 2L + 2w

 

32 = L + w

 

L = 32-w

 

 

BUT

Area = L*w

 

Area = (32-w)w  <---- area is now in terms of width w

 

 

To maximize the area, the MAX value occurs at the AVERAGE of the zeros of this parabola.

that is, to find where the max happens, find the zeros of the parabola, add them together, and divide by 2

 

The zeros are:

(32-w)w = 0

32-w = 0  OR w=0

w=32 or w=0

 

the average of these zeros is (32+0)/2 = 32/2 = 16

 

the max occurs when width w=16.

The length then must also be 16, as L = 32-w = 32-16 = 16

 

Since L=16 = w, it is a NOTICEABLE square.

Yes, this happens all the time.

 

To maximize the area of a rectangle, it turns into a square