miss guy is going to build a pen for her puppy. she has 50 feet of fencing to use. figure out some ways she can set up the pen and use all fencing. what are the dimension of the rectangular pen with the most space area for the puppy?
P = 2L + 2W
1 by 24
2 by 23
3 by 22
4 by 21
5 by 20
. . .
10 by 15
11 by 14
12 by 13
Area = length times width
To determine the dimensions of the rectangular pen with the maximum area, we can follow these steps:
1. Understand the problem: Miss Guy has 50 feet of fencing to use for her puppy's pen. We need to figure out the dimensions of the rectangular pen that will maximize the enclosed area.
2. Define the variables: Let's assume the length of the pen is L, and the width of the pen is W.
3. Determine the constraints: The total length of fencing used should be 50 feet. From the given information, we have 2L + 2W = 50.
4. Express the objective function: We want to maximize the area of the rectangular pen, which can be represented as Area = L × W.
5. Solve the problem: We can solve the constraints equation for one variable and substitute it into the objective function equation. Let's solve the constraints equation for L:
2L + 2W = 50
2L = 50 - 2W
L = 25 - W
Now, substitute this value of L into the objective function:
Area = (25 - W) × W
To maximize the area, we can take the derivative of the objective function with respect to W, set it equal to zero, and solve for W:
d(Area)/dW = -2W + 25
-2W + 25 = 0
2W = 25
W = 25/2
W = 12.5
Now, substitute this value of W back into the constraints equation to find the corresponding value of L:
2L + 2(12.5) = 50
2L + 25 = 50
2L = 50 - 25
L = 25/2
L = 12.5
Therefore, the dimensions of the rectangular pen with the maximum area are L = 12.5 feet and W = 12.5 feet.