6. 0/2 points details my notes scalc9 3.7.013...

6. 0/2 points details my notes scalc9 3.7.013. a farmer wants to fence an area of 37.5 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 lengths of the sides of the rectangular field be (in ft) in order to minimize the cost of the fence? smaller value 1580 x ft larger value 7,500,000 x ft enhanced feedback please try again, keeping in mind that the area of a rectangle with edges x and y is a = xy and the perimeter is p = 2x + 2y. also, consider the extra edge dividing the area in half. the length of the fence is the perimeter plus the extra edge. find a relationship between x and y, using the fact that the area is a constant. rewrite the amount of fencing as a function of one variable. use calculus to find the edges of the rectangle that minimize the amount of fencing.

Answer

# Explanation: ## Step1: Let the sides of the rectangle be $x$ and $y$. The area $A = xy=37500000$ (since $37.5$ million = $37500000$), so $y=\frac{37500000}{x}$. ## Step2: Assume the dividing - fence is parallel to side $x$. The length of the fence $L = 3x + 2y$. ## Step3: Substitute $y=\frac{37500000}{x}$ into the fence - length formula. $L(x)=3x + 2\times\frac{37500000}{x}=3x+\frac{75000000}{x}$. ## Step4: Take the derivative of $L(x)$. $L^\prime(x)=3-\frac{75000000}{x^{2}}$. ## Step5: Set the derivative equal to zero to find critical points. $3-\frac{75000000}{x^{2}} = 0$. Then $3x^{2}=75000000$, and $x^{2}=25000000$, so $x = 5000$. ## Step6: Find the value of $y$. Substitute $x = 5000$ into $y=\frac{37500000}{x}$, we get $y = 7500$. # Answer: smaller value: $5000$ ft larger value: $7500$ ft