Answer:
the dimensions of the corral that minimize the cost of the fencing are approximately 17.91 meters by 6.54 meters.
Explanation:
Let the length of the corral be x, and the width be y. We know that the area of the corral is 117, so:
xy = 117
We also know that the corral is divided by a fence into two sections. Let the length of one section be L, and the width be y. Then, the length of the other section will be x - L, and the width will still be y. The total fencing cost will be:
10(2x + 2y) + 6(L + 2y)
Simplifying this expression, we get:
20x + 32y + 6L
Using the area equation, we can rewrite this as:
20x + 32(117/x) + 6L
To minimize the cost, we need to take the derivative of this expression with respect to L, and set it equal to zero:
d/dL [20x + 32(117/x) + 6L] = 6 = 0
Solving for L, we get:
L = x/2
Therefore, the two sections of the corral will have equal length. Substituting this value of L into the expression for total fencing cost, we get:
20x + 32y + 3x
Simplifying this, we get:
23x + 32y
Using the area equation to substitute for y, we get:
23x + 32(117/x)
Taking the derivative of this expression with respect to x, and setting it equal to zero, we get:
23 - 3744/x^2 = 0
Solving for x, we get:
x = sqrt(3744/23) ≈ 17.91
Substituting this value of x into the area equation, we get:
y = 117/x ≈ 6.54
Therefore, the dimensions of the corral that minimize the cost of the fencing are approximately 17.91 meters by 6.54 meters.