6. 0/2 points details my notes scalc9 3.7.013...
6. 0/2 points details my notes scalc9 3.7.013. previous answers ask your teacher practice another 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 1586 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. need help? read it watch it
Answer
# Explanation:
## Step1: Set up variables and area equation
Let the sides of the rectangle be $x$ and $y$. The area $A = xy=37.5\times10^{6}$. So, $y=\frac{37.5\times10^{6}}{x}$.
## Step2: Set up the fencing - length equation
The length of the fence $L = 3x + 2y$ (assuming the extra fence is parallel to side $x$). Substitute $y=\frac{37.5\times10^{6}}{x}$ into the fencing - length equation: $L(x)=3x + 2\times\frac{37.5\times10^{6}}{x}=3x+\frac{75\times10^{6}}{x}$.
## Step3: Find the derivative of the fencing - length function
Differentiate $L(x)$ with respect to $x$. Using the power rule, if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. So, $L^\prime(x)=3-\frac{75\times10^{6}}{x^{2}}$.
## Step4: Set the derivative equal to zero and solve for $x$
Set $L^\prime(x)=0$. Then $3-\frac{75\times10^{6}}{x^{2}} = 0$. Add $\frac{75\times10^{6}}{x^{2}}$ to both sides: $3=\frac{75\times10^{6}}{x^{2}}$. Cross - multiply to get $3x^{2}=75\times10^{6}$. Then $x^{2}=25\times10^{6}$, and $x = 5000$ (we take the positive value since length cannot be negative).
## Step5: Find the value of $y$
Substitute $x = 5000$ into the area equation $y=\frac{37.5\times10^{6}}{x}$. So, $y=\frac{37.5\times10^{6}}{5000}=7500$.
# Answer:
smaller value: $5000$ ft
larger value: $7500$ ft