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 1 out of 2 $f(-4)=square$, meaning when the width of the rectangular area is $square$ ft, the area would be $square$ ft². this interpretation $checkmark$ in the context of the problem. $f(11.5)=square$, meaning when the width of the rectangular area is $square$ ft, the area would be $square$ ft². this interpretation $checkmark$ in the context of the problem. $f(16)=square$, meaning when the width of the rectangular area is $square$ ft, the area would be $square$ ft². this interpretation $checkmark$ in the context of the problem. based on the observations above, it is clear that an appropriate domain for the function is

Explanation:

Step1: Calculate $f(-4)$

Substitute $x = - 4$ into $f(x)=\frac{1}{2}x(30 - 2x)$.
\[

$$\begin{align*} f(-4)&=\frac{1}{2}\times(-4)\times(30-2\times(-4))\\ &=- 2\times(30 + 8)\\ &=-2\times38\\ &=-76 \end{align*}$$

\]
The width cannot be negative in the context of a real - world rectangle, so this interpretation is not valid.

Step2: Calculate $f(11.5)$

Substitute $x = 11.5$ into $f(x)=\frac{1}{2}x(30 - 2x)$.
\[

$$\begin{align*} f(11.5)&=\frac{1}{2}\times11.5\times(30-2\times11.5)\\ &=5.75\times(30 - 23)\\ &=5.75\times7\\ &=40.25 \end{align*}$$

\]
This means when the width of the rectangular area is $11.5$ ft, the area would be $40.25$ ft², and this interpretation is valid.

Step3: Calculate $f(16)$

Substitute $x = 16$ into $f(x)=\frac{1}{2}x(30 - 2x)$.
\[

$$\begin{align*} f(16)&=\frac{1}{2}\times16\times(30-2\times16)\\ &=8\times(30 - 32)\\ &=8\times(-2)\\ &=-16 \end{align*}$$

\]
The area cannot be negative in the context of a real - world rectangle, so this interpretation is not valid.

Step4: Determine the domain

Since the width $x\geq0$ and the length $\frac{1}{2}(30 - 2x)\geq0$.
Solve $\frac{1}{2}(30 - 2x)\geq0$:
\[

$$\begin{align*} 30-2x&\geq0\\ -2x&\geq - 30\\ x&\leq15 \end{align*}$$

\]
So the domain is $0\leq x\leq15$.

Answer:

$f(-4)=-76$, meaning when the width of the rectangular area is $-4$ ft, the area would be $-76$ ft². This interpretation is not valid.
$f(11.5)=40.25$, meaning when the width of the rectangular area is $11.5$ ft, the area would be $40.25$ ft². This interpretation is valid.
$f(16)=-16$, meaning when the width of the rectangular area is $16$ ft, the area would be $-16$ ft². This interpretation is not valid.
The domain is $0\leq x\leq15$.