a farmer wants to fence an area of 13.5 million square feet in a rectangular field and then divide it in…

a farmer wants to fence an area of 13.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. what should the sides of the rectangular field be (in ft) in order to minimize the cost of the fence? smaller value ft larger value ft

a farmer wants to fence an area of 13.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. what should the sides of the rectangular field be (in ft) in order to minimize the cost of the fence? smaller value ft larger value ft

Answer

Explanation:

Step1: Define variables

Let the length of the rectangle be (x) and the width be (y). The area (A = xy=13500000), so (y=\frac{13500000}{x}). The total length of the fence (L = 3x + 2y).

Step2: Substitute (y) into the fence - length formula

Substitute (y=\frac{13500000}{x}) into (L), we get (L(x)=3x + 2\times\frac{13500000}{x}=3x+\frac{27000000}{x}), (x>0).

Step3: Find the derivative of (L(x))

Using the power rule ((x^n)^\prime=nx^{n - 1}), (L^\prime(x)=3-\frac{27000000}{x^{2}}).

Step4: Set the derivative equal to zero and solve for (x)

Set (L^\prime(x) = 0), then (3-\frac{27000000}{x^{2}}=0). [ \begin{align*} 3&=\frac{27000000}{x^{2}}\ x^{2}& = 9000000\ x&=3000 \end{align*} ]

Step5: Check the second - derivative

Find the second - derivative (L^{\prime\prime}(x)=\frac{54000000}{x^{3}}). When (x = 3000), (L^{\prime\prime}(3000)=\frac{54000000}{3000^{3}}>0), so (L(x)) has a minimum at (x = 3000).

Step6: Find (y)

Substitute (x = 3000) into (y=\frac{13500000}{x}), we get (y = 4500).

Answer:

smaller value: (3000) ft larger value: (4500) ft