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A campground owner plans to enclose a rectangular field adjacent to a river.The owner wants the field to contain 180,000 square meters. What dimensions will use theleast amount of fencing?

Question

A campground owner plans to enclose a rectangular field adjacent to a river.
The owner wants the field to contain 180,000 square meters. What dimensions will use the
least amount of fencing?

check_circleAnswer
Step 1

A campground owner plans to enclose a rectangular field adjacent to a river. The owner wants the field to contain 180,000 square meters. What dimensions will use the least amount of fencing?

Let's assume the dimension of the field as length = L and breadth = B

Hence, area = LB = 180,000 sq m

Hence, B = 180,000 / L

Perimeter, P = 2 x (L + B) = 2 x (L + 180,000 / L)

Step 2

For the point of minima, following conditions must be satisfied:

  • First derivative should be zero
  • Second derivative should be positive
Step 3

Recall the famous rule of differentiation: d(xn) / dx = nxn-1

d{P(L)] / dL = 2 x (1 - 180,000 / L2) = 0

Hence, L2 = 180,000

Hence, L = square root of (180,000) = 424.26 meter

d2[P...

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Math

Calculus

Derivative

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