A rectangular dog run is to contain 864 ft ^2.
a. If the dog's owner must pay for fencing, what should be the dimensions of the run to minimize cost?
b. Suppose a neighbor has agreed to let the owner use an already constructed fence for one side of the run. What should the dimensions of the run be in this situation if the owner's cost is to be a minimum?
Can you explain how to do this?
Check the related question. I think you'll find that this problem has been answered many times, using different numbers.
85000
To solve this problem, we need to use optimization techniques to find the dimensions of the dog run that minimize cost.
a. If the dog owner must pay for fencing on all sides, including four sides, the dimensions of the dog run can be represented as length (L) and width (W). We want to minimize the cost, which is primarily determined by the amount of fence needed. The cost can be represented as the sum of the perimeters of all four sides.
The area of a rectangle is given by the formula A = L * W, and in this case, the area is given as 864 ft^2. So, we have LW = 864.
We're trying to minimize the cost, which is C = 2L + 2W (perimeter formula for a rectangle). We can rewrite this equation as C = 2L + 432/L (from the LW = 864 equation).
To minimize this cost, we take the derivative of C with respect to L, set it equal to zero, and solve for L. Let's differentiate:
dC/dL = 2 - 432/L^2
Setting it equal to zero:
2 - 432/L^2 = 0
432/L^2 = 2
L^2 = 432/2
L^2 = 216
L = √216
L ≈ 14.7 ft
So, the length of the dog run should be approximately 14.7 ft.
To find the width, we can substitute the value of L back into the LW = 864 equation:
14.7 * W = 864
W = 864/14.7
W ≈ 58.7 ft
Therefore, the dimensions of the dog run that minimize cost are approximately 14.7 ft by 58.7 ft.
b. In this case, one side of the dog run will use an already constructed fence. Let's say the already constructed fence is along the length (L) of the dog run, and the width (W) is the side where we need to minimize the cost.
Since one side is already fenced, we only need to fence three sides. The dog run's area can still be represented by LW = 864, but the cost equation changes to C = L + 2W.
Similarly, we differentiate dC/dW and set it equal to zero to find the minimum cost. We won't go through the entire calculation again, but after solving for W, you would find that W ≈ 34.9 ft.
Hence, with the already constructed fence along one side, the dimensions of the dog run that minimize cost are approximately 34.9 ft by 24.7 ft.