Quadratic Formula Word Problem Help?

Question

The question is as follows:

Kevin wants to create two adjacent pens for his sheep and goats. The pens will have the same dimensions, and can be built for $5/m. If he wants to have the maximum area for these pens, but can only spend $5400, what will the dimensions of each pen have to be?

I don't really have an idea what to do for this, my only thoughts would be:

5m = 5400
m = 5400/5
m = 1080
so we know it will have in total 1080 meters..
and then maby something like
2x(length) + 2x(width) = 1080
After this I cant think of anything, not even sure if what I just said is correct, help would be appreciated, thank you.

ireadlots · Answer

You are right that he has 1080 m of fencing to work with.

He wants 2 adjacent pens with the same dimensions.  Draw a rectangle.  Label one longer side x and one shorter side y.  Now draw a line parallel to side y inside the rectangle that divides it into 2 equal pens.

So, he needs 2x + 3y of fencing.  The total Area of his pens will be xy.

1080  = 2x + 3y
3y = 1080 - 2x
y = 360 - (2/3) x

Area = xy, so
Area (x) = x(360 - (2/3) x) = 360x - (2/3)x^2

To find the maximum, we set the first derivative to 0

A'(x) = 360-(4/3)x=0
x = 360*3/4 = 270

y = 360 - (2/3) x 
y = 360 - (2/3) 270 = 180

If you haven't done derivatives, find the vertex of the parabola y = 360x - (2/3)x^2

DWRead · Answer

Suppose each pen has length L and width W, and they share one common width-wise wall.
amount of material used = 4L + 3W meters
4L + 3W = 1080
L = 270 - 0.75W

area = 2LW 
= 2(270 - 0.75W)W
= 540W - 1.5W²

set first derivative to zero
540 - 3W = 0
W = 180 meters

L = 270 - 0.75·180 = 135 meters

Each pen is 135 meters long and 180 meters wide.

wiedyk · Answer

2L + 2W = (40 six) and L x W =(one hundred and twenty) take one equation and isolate one variable and then subsitute into the different equation. W = (40 six -2L)/2 = 23 -L L x (23-L) = one hundred and twenty ---> 23L -L^2 = one hundred and twenty (positioned L^2 on properly so its effective) 0=L^2 -23L + one hundred and twenty a = a million, b = -23, c = one hundred and twenty stick it into the quadratic formula and also you'll get 2 solutions for L (from the plus and minus from the formula) and then positioned the L value you got and positioned it back into between the origonal equations and clean up for W

Amar Soni · Answer

5m = 5400
 m = 5400/5
 m = 1080
 so we know it will have in total 1080 meters..
 and then maby something like
 2x(length) + 2x(width) = 1080
        2L +2W =1080
          L+W = 540  .................(i)
A =2L*2W
    =2L(1080-2L)
    = 2160L -4L^2..........(ii)
Differentiate
 dA/dL = 2160 -8L
For Max/Min    dA/dL = 0
              2160-8L =0
                      L =2160/8 =270m ..........Ans
                      W = 270m ............Ans

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Quadratic Formula Word Problem Help?

4 Answers