Question

. A field will be made in the shape of a rectangle with an area of 400 square meters. One side of the field is along a river and a fence will be built along the other three sides. A brick wall perpendicular to the river will be built to divide the field into two equal halves. the wall costs $20 per meter and the fence costs $10 per meter to build. what is the lowest possible cost to build such a field?

Answer 1

Answer:

The correct answer is $800.

Step-by-step explanation:

Let the length and width of the field be equal to l meters and b meters respectively and l > b.

Area of the field is given by l × b = 400 square meters.

The river is supposed to be along the longest side so that the price of fencing the other three sides is minimum. Thus the total perimeter of the fence is b+ b+ l = 2b+l.

Total cost for fencing the other sides of the field = $ 10 × (2b + l)

The wall is supposed to be perpendicular to the river and thus the length of the wall is b meters.

Total cost for the wall is $ 20 × b

Therefore, the total price for making the field is given by

C = 10 × (2b + l) + 20 × b

⇒ C = 40b + 10l

⇒ C = $\frac{16000}{l}$ + 10l

To minimize the cost we differentiate the cost with respect to l and equate it to zero.

$\frac{dC}{dl}$ = 0 = - $\frac{16000}{l^{2}}$ + 10

⇒ $l^{2}$ = 1600

⇒ l = 40 ; [ negative sign neglected as length cannot be negative ]

⇒ b = 10

The second order derivative of C is positive giving the minimum value of the cost.

Thus the minimum cost required to make the field is given by $800.