a Suppose you have 500 feet of fencing to enclose a rectangu

(a.) Suppose you have 500 feet of fencing to enclose a rectangular plot of land that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the maximum area?

(b.) A rectangular playground is fenced off and divided in two by another fence parallel to its width. If 900 feet of fencing is used, find the dimensions of the playground that will maximize the enclosed area. What is the maximum area?

(c.) A small car rental agency can rent every one of its 62 cars for $25 a day. For each $1 increase in rate, two fewer cars are rented. Find the rental amount that will maximize the agency\'s daily revenue. What is the maximum daily revenue?

Solution

a)

Let x = width of the pen
Let L = length of the pen
:
write an equation for the total fencing available (the perimeter)
2L + 2x = 500
Simplify, divide equation by 2
L + x = 250
L = (250-x)
:
The area = L * x, substitute (250-x) for x
A = x(250-x)
:
A = -x^2 + 250x; this is the variable expression for the area of the pen
:
You can calculate the max area using the axis of symmetry formula x=-b/(2a)
In this problem:
x = -250/(2*-1)
x = -250/-2

x = 125 is the width for max area
Note that would be a square: L = 250-x, right?
L = 250 - 125
L = 125 also
:
To find the actual area: substitute 125 for x in the equation
A = -(125^2) + 250(125)
A = -15625 + 31250
A = + 15625 sq/ft is the max area

b)

Let x = length of one side
Let L = length of the adjacent side:
:
An equation with 3 sides with a length of x, and two with a length of L
3x + 2L = 900
2L = 900 - 3x
L = (450 - 1.5x); divided both sides by 2
:
Area = L*x
Substitute (450 -1.5x) for L
A = x(450-1.5x)
A = 450x - 1.5x^2

Arrange like a quadratic
A = -1.5x^2 + 450x
:
Find the axis of symmetry, this is the value of x for max area
x = -b/(2a); in our equation a = -1.5; b = 450
x = -450/-3
x = +150 is dimension of x for max area
:
Find the max area; substitute 150 for x in the equation:
A = -1.5(150^2) + 450(150)
A = + 33750 sq ft is max area