Help me Solve this word problem??

Question

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Seven hundred and eighty feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum​ area?

1.The Smaller dimension is?___

2.The larger dimension is?___

3.The Maximum area is___ square feet?

Answer 1

Assume the lengths of the sides of the rectangle are a and b. The area A of the rectangle will then be

A = a b

WLOG, we can assume that the area is divided in two by cutting it parallel to the side having length a. This will mean that the total fence is

3 a + 2 b = 780 ⇒ b = (780 - 3 a) / 2 = 390 - 1.5 a.

Now the area can be expressed as a function of side length a alone,

A(a) = a (390 - 1.5 a) = 390 a - 1.5 a ².

The necessary condition for maximum A is that its derivative must vanish,

A^'(a₀) = 0 where

A^'(a) = 390 - 3 a. a₀ happens to be the value which maximized A. Once it is found, the corresponding length for the other side b₀ can be computed. The maximum area will be the product of a₀ and b₀.

Answer 2

With two variables (two dimensions) that are unknown, we can infer that solving this problem will require a system of equations, or two equations that both use our variables. For convenience, I'm going to call the smaller dimension, which can be thought of as the width of the fence, x. I will call the larger dimension, which can be thought of as the length of the fence, y. Now we need the two equations. Since we are told that we are looking for maximum area, an equation for area is definitely going to be one of them. We are also told that the total amount of fencing, or perimeter, is 780 feet. Perimeter is the second equation and must represent all sides of the fencing. There are two long sides (2y) and three short sides (3x), based on the description of how the fencing is placed. This is like a basic rectangle with the addition of a third section of fencing in the middle.

A=x*y

P=2y+3x or 780=2y+3x

Now, because we are looking for the maximum area, we have to find the derivative. The area of the rectangle can be thought of as a curve on a graph that increases and decreases depending on what the dimensions are. The area is at a maximum when the slope of this curve is zero. This point is also the vertex of the curve when the curve is in the shape of a downward facing parabola. To find the derivative of the area function, we need there to be just one variable, either x or y. It does not matter which you choose, so let's go with making A a function of just x. This means we need to replace y. To do that, we can solve the perimeter equation for y: 780=2y+3x

780-3x=2y

780/2-3x/2=y

Plugging this equation for y into A=x*y gives us: A=x(780/2-3x/2)

A=780x/2-3x²/2

Derivative of A: A=780x/2-3x²/2, where A is constant

2A=780x-3x²

0=780-6x Power Rule

x=130 feet

Now that we have x, we can use it and the perimeter equation to find y: 780=2y+3x

780=2y+3(130)

780-390=2y

390/2=y

y=195 feet

Finally, we can calculate the maximum area using our dimensions: A=x*y

A=(130)(195)

A=25,350 square feet