Question

2025fa-math-1314-65 - college algebra
hw score: 56.64%, 28.89 of 51 points
question 4, 4.4.9
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points: 0 of 10
a farmer has 9500 meters of fencing, and wants to enc a rectangular plot that borders on a river. if the farmer c not fence the side along the river, what is the largest ar that can be enclosed? (see the figure.)
the largest area that can be enclosed is

Explanation:

Step1: Define variables

Let the length of the side parallel to the river be $x$ and the other two equal - length sides perpendicular to the river be $y$. The total length of the fence is $x + 2y=9500$, so $x = 9500 - 2y$.

Step2: Express the area formula

The area $A$ of the rectangle is $A=xy$. Substitute $x = 9500 - 2y$ into the area formula, we get $A=(9500 - 2y)y=9500y-2y^{2}$.

Step3: Find the vertex of the quadratic function

The function $A(y)=- 2y^{2}+9500y$ is a quadratic function in the form $ay^{2}+by + c$ with $a=-2$, $b = 9500$, and $c = 0$. The $y$ - coordinate of the vertex of a quadratic function $f(y)=ay^{2}+by + c$ is given by $y=-\frac{b}{2a}$.
So $y=-\frac{9500}{2\times(-2)}=\frac{9500}{4}=2375$.

Step4: Find the value of $x$

Substitute $y = 2375$ into $x = 9500 - 2y$, we get $x=9500-2\times2375=9500 - 4750 = 4750$.

Step5: Calculate the maximum area

The maximum area $A = xy$. Substitute $x = 4750$ and $y = 2375$ into the area formula, $A=4750\times2375 = 11281250$ square meters.

Answer:

$11281250$ square meters