Question

malik has 30 feet of fence available to build a rectangular fenced - in area. if the width of the rectangle is $x$ feet, then the length would be $\frac{1}{2}(30 - 2x)$. a function to find the area, in square feet, of the fenced - in rectangle with width $x$ is given by $f(x)=\frac{1}{2}x(30 - 2x)$. find and interpret the given function values and determine an appropriate domain for the function.
answer attempt 2 out of 2
$f(-4)=-76$, meaning when the width of the rectangular area is $-4$ ft, the area would be $-76$ $ft^{2}$. this interpretation does not make sense in the context of the problem.
$f(11.5)=40.25$, meaning when the width of the rectangular area is $11.5$ ft, the area would be $40.25$ $ft^{2}$. this interpretation makes sense in the context of the problem.
$f(16)=-16$, meaning when the width of the rectangular area is $16$ ft, the area would be $-16$ $ft^{2}$. this interpretation does not make sense in the context of the problem.
based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Recall the function

The function for the area is $f(x)=\frac{1}{2}x(30 - 2x)=15x - x^{2}$.

Step2: Analyze domain conditions

The width $x\geq0$ and the length $\frac{1}{2}(30 - 2x)\geq0$. Solving $\frac{1}{2}(30 - 2x)\geq0$ gives $30-2x\geq0$, then $2x\leq30$ or $x\leq15$.

Answer:

$0\leq x\leq15$