Question

1. a lifeguard marks off a rectangular swimming area at a beach with 200 m of rope. what is the greatest area of water she can enclose if the rope only makes 3 sides of the rectangle? (no rope is needed along the shore side of the swimming area.)

Explanation:

Step1: Let the length parallel to the shore be $x$ and the other two equal - length sides perpendicular to the shore be $y$.

We know that the total length of the rope is 200 m, so $x + 2y=200$, which can be rewritten as $x = 200 - 2y$.

Step2: Express the area $A$ of the rectangle.

The area of a rectangle is $A=xy$. Substitute $x = 200 - 2y$ into the area formula, we get $A=(200 - 2y)y=200y-2y^{2}$.

Step3: Find the maximum of the area function.

This is a quadratic function $A(y)=-2y^{2}+200y$, where $a=-2$, $b = 200$, $c = 0$. For a quadratic function $f(y)=ay^{2}+by + c$, the vertex (which gives the maximum or minimum) occurs at $y=-\frac{b}{2a}$. Here, $y=-\frac{200}{2\times(-2)} = 50$.

Step4: Find the value of $x$.

Substitute $y = 50$ into $x = 200 - 2y$, we get $x=200-2\times50=100$.

Step5: Calculate the maximum area.

Substitute $x = 100$ and $y = 50$ into the area formula $A=xy$, we get $A = 100\times50=5000$ square - meters.

Answer:

$5000$ m²