John is fencing a larger rectangular pen and creating 3 smaller pens of equal area (2 lengths and 4 widths). John wants to maximize the area contained by the pens. John plans on using 1200m of fencing. What is the area and dimension of the 3 smaller pens? The larger pen?
Length ---- y
each of the 4 width --- x
so 2y + 4x = 1200
y = 600 - 2x
area of whole pen = xy = x(600-2x) = 600x - 2x^2
d(area...)/dx = 600 - 4x
= 0 for a max of area
4x = 600
x = 150
take over
if you have not had calculus
A = 600 x - 2x^2
x^2 - 300 x = -A/2
find vertex of parabola (the top since it opens down)
complete square
x^2 - 300 x + (300/2)^2 = -A/2 + 150^2
(x-150)^2 = -(1/2)(A - 45000)
so max at x =150, area = 45000
To find the dimensions and areas of the smaller pens and the larger pen, we can follow these steps:
1. Let's assume the length of the larger rectangular pen is L meters and the width is W meters.
2. We know that the perimeter of the larger pen is equal to the total amount of fencing John has, which is 1200 meters. So, we can create an equation using the perimeter formula: 2L + 2W = 1200.
3. Next, we want to express the length of the larger pen in terms of the width in order to maximize the area contained by the pens. We can solve the equation from step 2 for L:
2L = 1200 - 2W
L = (1200 - 2W) / 2
L = 600 - W
4. The area of the larger pen is calculated by multiplying the length and width: Area_LargerPen = L * W = (600 - W) * W.
5. To find the dimensions and area of the smaller pens, we divide the larger pen into three equal widths. So, each smaller pen will have a width of W/3 meters. Since the larger pen has a length of 600 - W meters, each smaller pen will also have a length of 600 - W meters.
6. Therefore, the area of each smaller pen is: Area_SmallerPen = (600 - W) * (W/3).
7. We want to find the value of W that maximizes the total area contained by the three smaller pens. To do this, we can use calculus to find the maximum value of the total area function.
8. The total area is calculated by multiplying the area of one smaller pen by 3: TotalArea = 3 * Area_SmallerPen = 3 * (600 - W) * (W/3).
9. Simplifying the total area equation gives us: TotalArea = (600 - W) * W.
10. To find the value of W that maximizes the total area, we can differentiate the total area equation with respect to W and set it equal to zero:
d(TotalArea)/dW = -W + 600 = 0
W = 600.
11. Now that we know the value of W, we can find the respective dimensions and areas of the smaller pens and the larger pen. The width of the smaller pens is W/3 = 600/3 = 200 meters, and the length is 600 - W = 600 - 600 = 0 meters.
12. Since the length of the smaller pens is 0 meters, their areas are also 0 square meters.
13. The area of the larger pen is: Area_LargerPen = L * W = (600 - W) * W = 0 * 600 = 0 square meters.
In conclusion, the dimensions of the three smaller pens are 200 meters in width and 0 meters in length. Therefore, the areas of the smaller pens are 0 square meters. The larger pen has an area of 0 square meters as well, given the dimensions obtained.