Question

A rectangular dog pen is to be made to enclose an area of 144 sq. ft. The pen is to be divided into 2 equal sections by a section of the fence. Assuming a total of 400 sq. feet of fence, find the dimensions of the pen that will require the least amonut of fencing material.
Ok so I started working the problem like this:
P=3x+2y
xy=144
y=144/x
400=3x+2(144/x)
then I worked it out to get:
3x^2-400x+288
Then I plugged this quadratic equation into the minimum vertex formula -b/2a
and got 66.67 as my answer.
Could someone please look at this and tell me if this is correct?
Thank you!!

Answer 1

Using the equation 400=3x+2(144/x)
you have actually imposed both conditions into your equation.
By setting it equal to 400, you are saying I will use all of the 400 feet, thereby nullifying the concept of finding a minimum length.

the question actually becomes a Calculus question.

let P be the perimeter.
P = 3x + 2y , as you had
but the area is supposed to be 144, xy=144, or y = 144/x (you had that)

then P = 3x + 2(144/x)
P = 3x + 288/x

this function is not a parabola, so the -b/2a part does not apply, you have to use derivatives

dP/dx = 3 - 288/x^2
= 0 for a minimum of P
so 288/x^2 = 3
.
.
x = 9.8 and subbing back
y = 14.7

so the minimum P is 3(9.8)+2(14.7) = 58.8
notice 9.8x14.7 = 144.06, close enough for 144

your answer of x=66.666 would give a y value of 2.16, a silly looking pen
your perimeter would be 204.32, clearly more in length than my answer, even though your area is 66.66666x2.16 = 144

Answer 2

Thank you very much!!

Answer 3

A tennis club offer two payment options
option 1. aaaaaaaaaa442 monthly fee plus $5/hour for court rental
option 2. No mothly fee but $8.50/hour for court rental.
Let x = hours per month for court time.
Write a mathematical model represnt toltal monthly cost, C in terms of x for the following.
option 1

option 2

Answer 4

Slove 5x+4y=12 for y

When graph, this equation would be line. By examing your answer to part a what is the slope and y intercept of this line