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!!
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
Thank you very much!!
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
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
To find the dimensions of the pen that will require the least amount of fencing material, you correctly started by defining the perimeter of the pen as P = 3x + 2y, where x and y are the dimensions of the pen.
Next, you correctly stated that the area of the pen is xy = 144.
Now, to address the total amount of fencing material available, you mentioned that it is 400 square feet. However, it's important to note that the question is asking for the total fence length, not the area. The length of the fence will be the perimeter, P.
So, using the formula for the perimeter P = 3x + 2y, and the given total fence length of 400, we can write the equation:
400 = 3x + 2y
From the earlier equation, xy = 144, we can solve for y:
y = 144/x
Substituting this value of y into the perimeter equation:
400 = 3x + 2(144/x)
Now, we need to find the value of x that minimizes the amount of fencing material. To do this, we can rewrite the equation as a quadratic equation:
3x^2 - 400x + 288 = 0
Now, you correctly mention applying the minimum vertex formula, which is the formula for finding the x-coordinate of the vertex of a quadratic equation. The formula is -b/2a, where a, b, and c are the coefficients of the quadratic equation.
In this case, the quadratic equation is 3x^2 - 400x + 288 = 0, so a = 3, b = -400, and c = 288. Plugging these values into the formula, we get:
x = -(-400) / (2 * 3) = 400 / 6 = 66.67
So, based on your calculations, 66.67 is the value of x that minimizes the amount of fencing material.
To find y, we can substitute this value of x back into the earlier equation y = 144/x:
y = 144/66.67 ≈ 2.16
Therefore, the dimensions of the pen that will require the least amount of fencing material are approximately 66.67 feet by 2.16 feet.