A rancher wants to build a rectangular pen with an area of 150 m^2?
a. Find an equation for the perimeter P in terms of W and L .
b. Use the given area to write an equation that relates W and L .
c. Find the pen dimensions that require the minimum amount of fencing.
Width =
Length =
as usual, P=2(W+L)
A = WL = 150
So, L = 150/W
P = 2(W + 150/W)
for minimum P,
dP/dW = 2 - 300/W^2 = 0, so
W = √150 = 5√6
a. To find an equation for the perimeter (P) in terms of the width (W) and length (L), we need to consider that the perimeter of a rectangle is given by the formula P = 2W + 2L.
b. Using the given area (A = 150 m^2), we can write an equation that relates the width and length of the rectangular pen. The formula for the area of a rectangle is A = W * L. Therefore, we have the equation A = W * L = 150.
c. To find the pen dimensions that require the minimum amount of fencing, we need to minimize the perimeter (P). Using the equation from part (a), we can substitute the expression for the perimeter in terms of width and length (P = 2W + 2L) into the equation from part (b) (A = W * L = 150). Simplifying, we get:
2W + 2L = P
W * L = 150
To minimize the amount of fencing, we need to express one variable in terms of the other and substitute it back into the equation for the perimeter. Let's solve the equation W * L = 150 for one variable:
L = 150 / W
Substituting L = 150 / W into the equation for the perimeter, we get:
2W + 2(150 / W) = P
To find the minimum amount of fencing required, we can differentiate the equation with respect to W and set it equal to zero:
dP/dW = 2 - (300 / W^2) = 0
Solving for W, we get:
2W = 300 / W^2
2W^3 = 300
W^3 = 150
W ≈ 5.84
Substituting W ≈ 5.84 back into the equation L = 150 / W, we get:
L ≈ 150 / 5.84 ≈ 25.68
Therefore, the dimensions that require the minimum amount of fencing are approximately:
Width ≈ 5.84 m
Length ≈ 25.68 m
To find the equation for the perimeter of the rectangular pen, we know that the perimeter of any rectangle is given by the formula P = 2W + 2L, where W represents the width and L represents the length of the rectangle. Hence, the equation for the perimeter in terms of W and L is P = 2W + 2L.
To write an equation that relates the width W and length L based on the given area of 150 m^2, we know that the area of a rectangle is given by the formula A = W * L. Therefore, the equation that relates W and L is W * L = 150.
To find the pen dimensions that require the minimum amount of fencing, we can use optimization techniques. We want to find the values of width W and length L that minimize the amount of fencing required, which is equal to the perimeter P. Since we already have the equation P = 2W + 2L, we need to substitute the equation W * L = 150 into the equation for P.
Substituting W * L = 150 into the perimeter equation gives us:
P = 2W + 2L
P = 2(W) + 2(L)
P = 2(W + L)
Now, we can substitute the value of P into the equation:
2(W + L) = 2W + 2L
2W + 2L = 2W + 2L
This means that no matter what values we choose for W and L, the perimeter will always be the same. In other words, there are no specific pen dimensions that require the minimum amount of fencing.