Please help me solve this.

Question

The owner of a play equipment store wants to construct a fence to enclose a rectangular outdoor storage area adjacent to the store, using part of the side of the store (which is 260 feet long) for part of one of the sides. (See the figure below.) There are 500 feet of fencing available to complete the job. Find the length of the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure.

Answer 1

l is the side parallel to the store.

w is the width, each of the two sides perpendicular to the store.

lw is area.

The fencing used is l + 2w = 500 (the store is the other length, so no fencing is used there.)

l = 500 - 2w Substitute that into the equation for area.

(500-2w)*w = area = 500w - 2w^2

That's an upside down parabola. Set the derivative = 0. That will be the maximum.

derivative = 500 - 4w = 0

500 = 4w

125 = w

l = 500 - 2w = 500 - 2(125) = 500 - 250 = 250.

250 of the store wall and 250 feet of fence make the lengths.

125 feet of fence makes each of the two sides perpendicular to the wall.

250 + 125 + 125 = 500 feet of fence used.

Answer 2

Let's denote the length of the sides parallel to the store by x and the length of the sides perpendicular to the store by y. Then we have the following diagram:

_______________________

| x | |

| | |

| | |

|___________|___________|

y 260-x

We want to maximize the area of the enclosure, which is given by A = xy. We have two constraints: the amount of fencing available and the fact that the enclosure is rectangular.

The amount of fencing we have available is 500 feet. One of the sides parallel to the store is already accounted for by the store itself, so we need to fence three sides with a total length of 500 - 260 = 240 feet. This means that 2x + y = 240.

The fact that the enclosure is rectangular means that opposite sides have equal length. So we have x = 260 - y.

Substituting x = 260 - y into the equation 2x + y = 240, we get:

2(260 - y) + y = 240

Simplifying this equation, we get:

520 - y = 240

y = 280

Therefore, x = 260 - y = 260 - 280 = -20. This doesn't make sense, so we know that we made a mistake. The mistake is in assuming that opposite sides have equal length. In fact, we need to fence two sides with a length of x and two sides with a length of y, so we have the equation:

2x + 2y = 500 - 260 = 240

Simplifying this equation, we get:

x + y = 120

Now we can use the fact that A = xy to write the area in terms of one variable. Solving for y, we get:

y = 120 - x

Substituting this into the equation A = xy, we get:

A = x(120 - x)

Expanding and simplifying, we get:

A = 120x - x^2

This is a quadratic function of x, which has a maximum value at the vertex of the parabola. The x-coordinate of the vertex is given by x = -b/2a, where a = -1 and b = 120. So we have:

x = -b/2a = -120/(-2) = 60

Therefore, y = 120 - x = 60.

The length of the sides parallel to the store is 60 feet, and the length of the sides perpendicular to the store is also 60 feet. The maximum area of the enclosure is A = xy = 60 * 60 = 3600 square feet.