The management of a large store has 1600 feet of fencing to fence in a rectangular storage yard using the building as one side of the yard. If the fencing is used for the remaining 3 sides, find the area of the largest yard possible.
320,000
To find the area of the largest yard possible, we need to determine the dimensions of the rectangular yard that will maximize the area. Let's break down the problem step by step:
Step 1: Understand the problem.
We have 1600 feet of fencing to enclose three sides of a rectangular yard, with the fourth side being the building. We need to maximize the area of the yard.
Step 2: Identify the variables.
Let's assume the width of the yard is 'w' feet and the length (parallel to the building) is 'l' feet.
Step 3: Formulate the equations.
The equations we can form from the given information are:
1. The perimeter equation: 2w + l = 1600 feet (since we have to fence in three sides)
2. The area equation: A = w * l
Step 4: Solve the equations.
Solve the perimeter equation for one of the variables and substitute it in the area equation to get a single-variable expression for the area.
From equation 1, we can rewrite it as l = 1600 - 2w.
Substituting this in the area equation, we get A = w * (1600 - 2w).
Step 5: Find the maximum of the area.
To find the maximum area, we need to find the value of 'w' that maximizes the expression A = w * (1600 - 2w). We can do this by finding the critical points or using calculus.
Considering the quadratic expression A = w * (1600 - 2w), we can find the maximum when the derivative is equal to zero.
Differentiating A with respect to w, we get dA/dw = 1600 - 4w.
Setting dA/dw = 0, we have 1600 - 4w = 0.
Solving this equation, we find w = 400 feet.
Step 6: Find the length and area.
Substitute the value of 'w' back into the perimeter equation to find the length:
2w + l = 1600
2 * 400 + l = 1600
800 + l = 1600
l = 800 feet
Calculate the area using the dimensions obtained:
A = w * l
A = 400 * 800 = 320,000 square feet.
Therefore, the largest possible area for the rectangular yard is 320,000 square feet.