Question

A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be enclosed with a protective fence. If there are 36 km of fence available, find the dimension of the rectangle to maximize the area of the sanctuary. The area of the sanctuary is maximized when _____ km of fencing is used for each side perpendicular to the riverbank and _____ km of fencing is used parallel to the riverbank.

Answer 1

To maximize the area of the sanctuary, 9 km of fencing is used for each side perpendicular to the riverbank and 18 km of fencing is used parallel to the riverbank.

Step-by-step explanation:

To maximize the area of the sanctuary, we need to find the dimensions of the rectangle that use up all 36 km of available fence. Let's assume the side perpendicular to the riverbank is x km long and the side parallel to the riverbank is y km long. The remaining side opposite the riverbank will also be x km long.

The perimeter equation for the rectangle is:

2x + y = 36

Solving for x, we get:

x = (36 - y)/2

The area of the sanctuary is given by:

Area = x * y = ((36 - y)/2) * y = -0.5y^2 + 18y

To find the maximum area, we take the derivative of the area equation with respect to y and set it equal to 0:

d(Area)/dy = -y + 18 = 0

Solving for y, we get y = 18 km.

Substituting this value back into the perimeter equation, we find x = 9 km.

Therefore, the area of the sanctuary is maximized when 9 km of fencing is used for each side perpendicular to the riverbank and 18 km of fencing is used parallel to the riverbank.