A contractor is to fence off a rectangular field along a straight river, the side along the river requiring no fence.
What is the least amount of fencing needed to fence of 30,000 meters squared?
perimeter = L + 2 W = p
L W = 30,000
so
L = 30,000/W
then
p = 30,000/W + 2 W
dp/dW = -30,000/W^2 + 2
for in or max, dp/dW = 0
30,000 = 2 W^2
W^2 = 15,000
W = 122.5
then L = 30,000/122.5 = 244.9
above gives p = 244.9+2*122.5 = 490
If you do not know derivatives then
p = 30,000/W + 2 W
2 W^2 - Wp +30,000 = 0
W^2 - (p/2)W = -15,000
W^2 - (p/2)W + p^2/16 = -15,000 + p^2/16
(W-p/4)(W-p/4) = 1/16( p^2 - 240,000)
that is vertex at W = p/4 and p = sqrt(240,000) or 490 sure enough
To determine the least amount of fencing needed to fence off a rectangular field with a given area, we need to find the dimensions of the rectangle that would maximize the area while minimizing the perimeter.
Let's break down the problem step by step:
1. Let's assume the length of the rectangle is represented by 'L' and the width is represented by 'W'.
2. Since the side along the river requires no fence, the rectangle will have three sides that require fencing (top, bottom, and one side).
3. The area of a rectangle is given by the formula: A = L * W, where A represents the area.
4. We are given that the area of the field is 30,000 m², so we can write the equation: 30,000 = L * W.
5. We need to minimize the perimeter, which is the sum of the lengths of all the sides.
6. In this case, the perimeter, P, is given by P = L + L + W = 2L + W.
Now, to minimize the perimeter, we can use the concept of derivatives. To find the minimum value of the perimeter, we need to find the stationary point (where the derivative equals zero) or the endpoints (where L or W equals zero, which is not applicable in this scenario).
Taking the derivative of P with respect to L, we get:
dP/dL = 2.
Setting the derivative equal to zero and solving for L, we find that L = 0.
Since L cannot be zero in this context, we can conclude that L is not relevant in finding the minimum amount of fencing required.
Now, let's solve for W using the given equation A = L * W:
30,000 = W * L.
Substituting L = 30,000 / W, we get:
30,000 = W * (30,000 / W).
Simplifying the equation, we find:
W² = 30,000.
Taking the square root of both sides, we get:
W = √30,000 ≈ 173.21.
So, the width of the rectangular field is approximately 173.21 meters.
Since the length (L) is not relevant, we can ignore it.
Now, let's calculate the minimum amount of fencing needed:
Perimeter (P) = 2L + W = 2(0) + 173.21 = 173.21 meters.
Therefore, the least amount of fencing needed to fence off a rectangular field with an area of 30,000 m² is approximately 173.21 meters.