You have 700 ft of fencing to make a pen for hogs. If you have a river on one side of your property, what are the dimensions (in ft) of the rectangular pen that maximize the area?
shorter side
longer side
I suggest you try different dimensions. Which will give you the greatest area?
400 feet along the river; 400 feet of fencing parallel to the river, 150 feet of fencing at each end
or
600 feet along the river, 600 feet of fencing parallel to the river, 10 feet of fencing at each end
or
?
If you play around with it, you will see that the maximum area in such problems is always attained when the fencing is divided equally among lengths and widths. Here, you have one length and two widths, so if 350 ft of fencing is allocated to each set, then the pen will be
350 x 175
for maximum area
I think I can mathematically prove it using L'Hopitals rule.
To find the dimensions of the rectangular pen that maximize the area, we can use the concept of a quadratic equation. Let's assume the shorter side of the rectangle is "x" feet. In that case, the longer side will be "700 - 2x" feet (since we have a river on one side, and we need to subtract the length of that side from the total length of the fence).
The formula for the area of a rectangle is length times width, so the area (A) of the rectangular pen can be expressed as:
A = x * (700 - 2x)
To maximize the area, we need to find the value of "x" that maximizes the quadratic equation. One way to do this is by finding the vertex of the quadratic equation.
The vertex of a quadratic equation in the form of "ax^2 + bx + c" can be found using the formula: x = -b/2a. In our case, the equation is "A = -2x^2 + 700x".
By comparing this equation to the ax^2 + bx + c form, we can see that a = -2, b = 700, and c = 0. Applying the formula for the x-coordinate of the vertex, we have:
x = -b/2a
x = -700 / (2 * -2)
x = -700 / -4
x = 175
So, the value of "x" that maximizes the area is 175 feet. To find the longer side of the rectangle, we substitute this value back into our equation:
longer side = 700 - 2x
longer side = 700 - 2 * 175
longer side = 700 - 350
longer side = 350
Therefore, the dimensions (in feet) of the rectangular pen that maximize the area are:
Shorter side: 175 feet
Longer side: 350 feet