Farmer Jones has 480 feet of fence. She wishes to construct a rectangular pen divided into five separate pens, with one of the pens twice as large as each of the other four (see figure). She must use part of her 480 feet of fencing material to make the partitions. What should the dimensions of the combined pens be to maximize the area enclosed?
The horizontal dimension should be _______ feet.
The vertical dimension should be ______ feet.
The resulting total area is square _______ feet.
h=60
L=30
To maximize the area enclosed by the rectangular pen, we need to find the dimensions that will use up all of the available fence material of 480 feet.
Let's assume the horizontal dimension of the combined pens is x feet. Since there are five separate pens, each partition will have a width of x/5 feet.
Now let's find the length of each partition:
- The large central pen will have a length of 2x/5 feet (twice as large as the other four).
- Each of the remaining four pens will have a length of x/5 feet.
Now let's calculate the total amount of fence material used for the partitions:
- The large central pen will have two lengths, so the amount of fence used for this pen is 2 * (2x/5) = 4x/5 feet.
- The four smaller pens will have four lengths, so the amount of fence used for these pens is 4 * (x/5) = 4x/5 feet.
Since Farmer Jones has 480 feet of fence material, the total amount of fence used for the partitions is (4x/5) + (4x/5) = 8x/5 feet.
To find the remaining fence used for the outer enclosure of the combined pens, we subtract this from the total fence material available:
480 - (8x/5) = (240 - 8x/5) feet.
The outer enclosure will consist of two lengths and two widths, so we multiply these remaining fence lengths by 2:
2 * (240 - 8x/5) = 480 - (16x/5) feet.
Now, let's find the dimensions that maximize the area enclosed. The area is given by length * width, so:
Area = (2x/5) * (240 - 8x/5)
To simplify the expression, we can cancel out the 5 in the denominator:
Area = (2x/5) * (48 - 8x)
To maximize the area, we can differentiate the expression with respect to x and set it equal to zero:
d(Area)/dx = 0
Simplifying, we get:
d(Area)/dx = 96/25 - 4x/5 = 0
Solving this equation for x gives us:
96/25 = 4x/5
96 * 5 = 25 * 4x
480 = 100x
x = 480/100 = 4.8 feet
Now that we have found the value of x, we can substitute it back into the expression for the length and width:
Length = 2x/5 = 2 * 4.8/5 = 9.6/5 = 1.92 feet
Width = 240 - 8x/5 = 240 - 8 * 4.8/5 = 240 - 38.4/5 = 240 - 7.68 = 232.32 feet
Therefore, the dimensions of the combined pens that will maximize the area enclosed are:
- The horizontal dimension is 4.8 feet.
- The vertical dimension is 232.32 feet.
- The resulting total area is (2x/5) * (240 - 8x/5) = 1.92 * 232.32 square feet.