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?
Area= R*W where road fence is R, W is the length perpendicular to the Road.
Cost= 34R+16(R+2W)= 34Area/W+ 16(Area/W+2W)
dCost/dw= -34*100/W^2+ 16(-100/W^2+2)=0
0=-3400-1600+32W^2
W= sqrt (5000/32)
L= 100/W
check my math.
let there be length a and width b, with side a along the road.
b = 100/a
cost is a*34 + a*16 + 2*100/a * 16
c = 50a + 200/a
c' = 50 - 200/a^2
c' = 0 when a = 2
so, the minimum cost is 100 + 100 = 200
a 2' wide pen? Is he housing gerbils?
My bad - bobpursley is correct. The road length is 8, width is 12.5
using my notation,
c = 50a + 3200/a
To minimize the cost of fencing, we need to find the dimensions of the rectangular pen with an area of 100 square feet that result in the minimum cost. Let's assume the length of the pen is L feet and the width is W feet.
The area of a rectangle is given by the formula: A = L * W. Since we know the area is 100 square feet, we can write the equation: L * W = 100.
To find the minimum cost, we need to consider the cost of the fence along the road and the cost of the other three sides.
The fence along the road costs $34 per foot and the other three sides cost $16 per foot. The cost of the fence along the road is determined by the length, so it is equal to 34 * L. The cost of the other three sides is determined by the perimeter (pen's width plus twice the pen's length), so it is equal to 16 * (W + 2L).
To find the minimum cost, we can write a cost function C(L, W) that represents the total cost in terms of L and W:
C(L, W) = 34 * L + 16 * (W + 2L)
Now, we need to substitute the area equation into the cost function to have a cost function in terms of a single variable.
Rewriting the area equation, we have W = 100 / L. Substituting this into the cost function, we get:
C(L) = 34 * L + 16 * (100 / L + 2L)
Now, we have the cost function expressed in terms of a single variable L. To find the minimum cost, we will differentiate the cost function with respect to L, set it to zero, and solve for L.
C'(L) = 34 - 16 * (100 / (L^2) + 2) = 0
Multiplying through by L^2, we get:
34L^2 - 16 * (100 + 2L^3) = 0
Simplifying, we have:
34L^2 - 3200 - 32L^3 = 0
Now, we need to solve this equation to find the value of L that gives the minimum cost. However, this equation is complex to solve analytically. We can use numerical methods like Newton's method or computer software to find the solution.
Once we find the value of L that minimizes the cost, we can substitute it back into the area equation W = 100 / L to find the corresponding value of W.
Finally, we can plug L and W into the cost function C(L, W) = 34L + 16(W + 2L) to calculate the minimum cost.
Therefore, to find the dimensions for the pen and the minimum cost, we need to solve the equation 34L^2 - 3200 - 32L^3 = 0 to get the value of L, then calculate W using W = 100 / L, and finally plug the values of L and W into the cost function C(L, W) = 34L + 16(W + 2L) to find the minimum cost.