A rectangular dog pen is to be sectioned off into two separate areas and made to enclose area of 225 feet. Find the dimensions of the pen that will require the least amount of fencing material.
To find the dimensions of the dog pen that will require the least amount of fencing material, we can use the concept of optimization.
Let's denote the length of one of the sides of the rectangular dog pen as x. If we divide the pen into two equal sections, the width of each section will be 225/x since the total area is 225 square feet.
Now, we can calculate the amount of fencing material needed. Each section will have a length of x and a width of 225/x. For each section, we need two lengths and two widths of fencing material. Therefore, the total amount of fencing material needed is given by the expression:
F(x) = 2x + 2(225/x)
To find the value of x that minimizes F(x), we can take the derivative of F(x) with respect to x and set it equal to zero. Let's do that:
F'(x) = 2 - 450/x^2
Setting F'(x) equal to zero:
2 - 450/x^2 = 0
Solving for x:
450/x^2 = 2
450 = 2x^2
225 = x^2
x = √225
x = 15
Therefore, the dimensions of the dog pen that will require the least amount of fencing material are 15 feet by 15 feet.