Math 1111 homework

Question

My friend has $1280 to spend on a fence for her rectangular garden. She wants to use cedar fencing which costs $22/meter on one side, and cheaper metal fencing which costs $10/meter for the other three sides.

What are the dimensions of the garden with the largest area she can enclose?

length for cedar side =  meter

width for other side =  meter

What is the largest area that can be enclosed?

area =  meter-squared

If necessary, round your answers accurate to two decimal places.

Answer 1

Hi, Austin!

The preview on my end of this video isn't playing any audio, so I'll type out a text version as well so you don't have to follow along with a silent video.

Basically, this is an optimization problem, using your budget as your primary constraint. Before you do anything, let's break down this problem.

Since our garden is rectangular, there are going to be two elements of our area equation: length and width. Our cedar side, is, again our length.

If we add 1 meter of width to our garden, we will be adding 1 meter of metal fence on our top side and one meter of metal fence to our bottom side. So each meter of width will entail 2 meters of metal fence, which costs $10/meter. Therefore, 1 meter of width will cost $20 in total.

We can do something similar for length. Each meter of length will have 1 meter of metal on the left, and 1 meter of cedar on the right, costing $32 in total.

It's important to remember that each meter of length that we add to the fence is 32 dollars we cannot use to buy width.

Our total budget is $1280. So to represent "How much money can we spend on width after buying X amount of meters of length", we can use this equation.

Money for width = 1280 - 32X

And since width costs 20 dollars per meter, the amount of width we have will be defined as:

(1280-32x)/20

So taking our width above and multiplying it by our length, X, we get this equation:

Area = x * (1280-32x)/20.

This simplifies down to 64x-1.6x^2

If you graph this, you get a parabola with a clear-cut maximum for area.

To get the maximum for this parabola, you can use 1 of two methods. I used a derivative.

If f(x) = 64x-1.6x^2, then f'(x) = 64-3.2x.

Remember that when the derivative of an equation equals zero, its slope equals zero, meaning that (on a parabola), it's either reached a minimum or a maximum. In our case, that's our maximum. So let's solve for X when f'(x) equals zero.

0 = 64-3.2x

x = 20.

Alternatively, you could figure out the roots of this equation and take the midpoint between them.

64x - 1.6x^2 = x * (-1.6x + 64)

One of our roots, represented by "x", is 0. Our other, can be observed by solving that side for x.

0 = -1.6x + 64

x = 40.

The midpoint between 0 and 40 is, again, 20.

As such, the length of our garden will be 20 meters.

Plugging this back into our equation for width, this means our width will be 32 meters, and our area is 640 square meters.

So, to sum up:

Length: 20 meters

Width: 32 meters

Area: 640 meters squared.

Hope this helps!