Answer:
- Length of the fence along the front part: approximately 179 meters
- Length of the fence for the other three sides: approximately 358 meters (2x) and 446.93 meters (y)
- Minimum cost of the fence: approximately $13419.3
Explanation:
Let's denote the length of the front part of the land as x and the width as y. We know that the area of the land is 80000 m², so we can write the equation:
xy = 80000
The cost of the fence along the front part is $30 per meter, so the cost for that part of the fence would be 30x. The cost for the other three sides would be $10 per meter, so the cost for those sides would be 10(2x + y) since there are two sides of length x and one side of length y.
To minimize the cost, we need to express the cost in terms of a single variable and then find its minimum value.
Let's solve for y in terms of x using the equation for the area:
y = 80000 / x
Substituting this into the cost equation, we get:
Cost = 30x + 10(2x + 80000 / x)
Simplifying this expression:
Cost = 30x + 20x + 1600000 / x
To find the minimum cost, we take the derivative of the cost function with respect to x, set it equal to zero, and solve for x:
d(Cost)/dx = 30 + 20 - 1600000 / x² = 0
50 - 1600000 / x² = 0
1600000 / x² = 50
x² = 1600000 / 50
x² = 32000
x = sqrt(32000) ≈ 178.89
Since x represents the length of the fence along the front part, which must be a whole number, we can round it to the nearest whole number:
x ≈ 179 meters
Now we can calculate the width using the area equation:
y = 80000 / x
y = 80000 / 179 ≈ 446.93 meters
To find the minimum cost, we substitute the values of x and y into the cost equation:
Cost = 30x + 10(2x + y)
Cost = 30(179) + 10(2(179) + 446.93)
Cost = 5370 + 10(358 + 446.93)
Cost = 5370 + 10(804.93)
Cost = 5370 + 8049.3
Cost ≈ $13419.3
Therefore, the minimum cost of the fence is approximately $13419.3.