A farmer wishes to enclose a rectangular pen with area 100 square feet next to a road. The fence along the road is to be reinforced and costs $34 per foot. Fencing that coast $16 per foot can be used for the other three sides. What dimensions for the pen will minimize the cost to the farmer. What is the minimum cost?
To find the dimensions that will minimize the cost, we need to express the cost as a function of one variable and then find the minimum of that function. Let's assume the length of the rectangular pen is x feet, and the width is y feet.
1. Express the cost as a function of x:
- The fence along the road has a cost of $34 per foot, and since the length is x feet, the cost of that side is $34x.
- The other two sides have a cost of $16 per foot. Each side has a length of y, so the cost for both sides combined is $16y + $16y = $32y.
- Total cost function, C(x, y) = $34x + $32y.
2. Express the width y in terms of x and the area of the pen:
- The area of the rectangular pen is given as 100 square feet: xy = 100.
- Rearrange the equation to obtain y = 100/x.
3. Substitute y = 100/x into the cost function:
- C(x) = $34x + $32(100/x) = $34x + $3200/x.
4. Find the derivative of C(x) with respect to x:
- C'(x) = 34 - 3200/x^2.
5. Set C'(x) = 0 and solve for x:
- 34 - 3200/x^2 = 0.
- Cross-multiply: 34x^2 = 3200.
- Divide by 34: x^2 = 3200/34 = 94.12.
- Take the square root: x = sqrt(94.12) = 9.70 (approximately).
6. Test the critical points by using the second derivative test:
- C''(x) = 6400/x^3.
- If C''(x) > 0, the critical point is a local minimum. Otherwise, it is a local maximum.
- Plug in x = 9.70: C''(9.70) = 6400/(9.70)^3 ≈ 7.29.
- Since C''(9.70) > 0, x = 9.70 is a local minimum.
7. Find the corresponding width y:
- Substitute x = 9.70 into the equation y = 100/x: y = 100/9.70 ≈ 10.31.
Therefore, the dimensions that minimize the cost are approximately 9.70 feet by 10.31 feet. To find the minimum cost, substitute these values back into the cost function:
C(9.70, 10.31) = $34(9.70) + $32(10.31) ≈ $901.46.
Hence, the minimum cost to the farmer is approximately $901.46.