12.2 Precalc Homework

Question

I would like to create a rectangular vegetable patch. The fencing for the east and west sides costs $4 per foot, and the fencing for the north and south sides costs only $2 per foot. I have a budget of $96 for the project. What are the dimensions of the vegetable patch with the largest area I can enclose?

north and south sides 12 ft

 east and west side __ ft

Answer 1

Hi Jenna,

Let's introduce some variables to describe the side lengths of the vegetable patch.

x = length of the north/ south side

y = length of the east/ west side

You have a total of $96 to spend. Each foot of north/ south fencing (x) contributes $2. Each foot of east/ west fencing (y) contributes $4. We can write a cost equation to describe this:

2x + 4y = 96

The area of this vegetable patch is:

A = x*y

To get this into one variable, solve the cost equation for y.

2x + 4y = 96

4y = 96 - 2x

y = 24 - 1/2x

Now you can write the area function as:

A = x*(24 - 1/2x)

A = 24x - 1/2x²

This is an upside-down parabola. Since you want the maximum area, find the vertex using the equation:

x_max = -b/2a

= -(24)/(2*(-1/2))

= 24

So the max area occurs when x = 24. We can plug this into the cost equation where we solved for y.

y = 24 - 1/2*24

y = 12

So you want your north/ south fences to be 24 feet long, and your east/ west fences to be 12 feet long.

Hope this helps!