Question

Sue wants to put rectangular garden on her property using 90 meters of fencing. There is a ns through her property, so she decides to increase the size of the garden by ne side of the rectangle. (Fencing is then needed only on the other three river that runs through her property, so she using the river as one side of the rectangle. (ren sides). Let x represent the length of the side of the rectangle parallel

a. Express the garden's area as a function of x.
b. Using your graphing calculator, determine the maximum area that Sue will be able to enclose for her garden and the dimensions of that area. (Hint: A good viewing window might be (-40, 100; 0, 1100])

Answer 1

a) A(x) = 90*x/2 - x²/2

b) A(max) = 1012.5 m²

Explanation:

L = 90 meters of fencing

Rectangular area is

A(r) = x*y . where x and y are the sides of the rectangle

the perimeter is ( we are going to fence only 3 sides, then)

x + 2*y = 90 or . y = ( 90 - x ) /2

Area as a function of x is:

A(x) = x * ( 90 - x)/2

A(x) = 90*x/2 - x²/2

Tacking derivatives on both sides of the equation:

A´(x) =45 - 2*x/2 A´(x) =45 - x

A´(x) = 0 . 45 - x = 0 . x = 45 . meters

and . y = ( 90 - x ) 2

y = ( 90- 45 )/2

y = 22.5 meters

A(max) = 45*22.5 m²

A(max) = 1012.5 m²

If we get the second derivative of A(x) . A"(x) = - 1 A"(x) < 0

Then A(x) has a maximum for x = 45