Question

250 square foot section with a fence that costs $1.50 per yard. What dimensions would you use to minimize the cost?

In your own words, please post a response

Answer 1

Lets make the dimensions of this area x for length, and y for width. Since the area is 250 square feet, we have:
\[xy = 250 \Rightarrow y = \frac{250}{x}\]
At the same time, the perimeter is going to be:
\[2(x+y) \Rightarrow 2(x+\frac{260}{x})\]
To minimize cost, i want to minimize the perimeter, so im going to take the derivative and set it equal to zero:
\[P = 2x+\frac{500}{x} \Rightarrow 0 = 2-\frac{500}{x^2}\]
Now we just solve for x:
\[0 = 2-\frac{500}{x^2} \Rightarrow 0 = 2x^2-500 \Rightarrow 2x^2 = 500 \Rightarrow x^2 = 250 \Rightarrow x = 5\sqrt{10}\]
The last part is figuring out what y is:
\[y = \frac{250}{x} = \frac{250}{5\sqrt{10}} = \frac{50}{\sqrt{10}} = \frac{50\sqrt{10}}{10} = 5\sqrt{10}\]

Answer 2

Joe thank you