Suppose you have enough material for 60 feet of fencing. With this material you want to build the largest rectangular enclosure possible. What are the dimensions of the largest possible enclosure, and what is its area?
I'd build a square enclosure with 15 feet on each side.
Instead of 60 feet, let’s say you have 4x feet of fencing, where x is any number. What, in terms of x, should the dimensions of the largest rectangular enclosure be, whose perimeter is 4x feet? What would the area be?
To determine the dimensions of the largest possible rectangular enclosure and its area, we can follow these steps:
Step 1: Understand the problem
We are given a fixed amount of material for fencing, which is 60 feet in this case. We need to determine the dimensions (length and width) of a rectangular enclosure that uses all of the given material and has the largest possible area.
Step 2: Define variables
Let's assume the length of the rectangular enclosure is 'L' and the width is 'W'. We want to find the values of L and W that maximize the area of the enclosure.
Step 3: Identify constraints and formulate an equation
Since we have a fixed amount of fencing material, the perimeter of the enclosure (P) must be equal to the given amount of material, which is 60 feet. We can write this as an equation:
P = 2L + 2W = 60
Step 4: Solve the equation for one variable
Let's solve the equation for one variable (L or W) in terms of the other variable. We can rearrange the equation as follows:
2L + 2W = 60
2L = 60 - 2W
L = 30 - W
Step 5: Substitute the expression into the objective function
The objective function in this case is the area of the rectangular enclosure, which is given by the product of length and width (A = L * W):
A = (30 - W) * W
Step 6: Optimize the objective function
To maximize the area, we need to find the value of W that gives us the largest possible value for A. We can do this by finding the vertex of the quadratic equation A = -(W^2 - 30W).
Step 7: Find the vertex
The vertex of a quadratic equation in the form A = ax^2 + bx + c, where a, b, and c are constants, can be found using the formula x = -b / (2a).
In our case, a = -1, b = 30, and c = 0. Plugging these values into the formula, we get:
W_vertex = -(30) / (2 * (-1)) = 15
Step 8: Calculate the dimensions and area
Now that we have the value of W, we can substitute it back into the equation L = 30 - W to find the corresponding value of L:
L = 30 - 15 = 15
So, the dimensions of the largest possible rectangular enclosure are L = 15 feet and W = 15 feet. The area of the enclosure, A, can be found by substituting the values of L and W into the area formula:
A = L * W = 15 * 15 = 225 square feet.
Therefore, the dimensions of the largest possible enclosure are 15 feet by 15 feet, and its area is 225 square feet.