Question

suppose that the number of worker-hours required to distribute new telephone books to $x\\%$ of the households in a certain rural community is given by the function
$$w(x) = \frac{800x}{700 - x}.$$
(a) what is the domain of the function $w$? $(0,100)$ (give the domain in interval notation. if the answer includes more than one interval write the intervals separated by the \union\ symbol, $\cup$. if needed enter $\infty$ as infinity and $-\infty$ as -infinity.)
(b) for what values of $x$ does $w(x)$ have a practical interpretation in this context? $0 \leq x \leq 100$
(c) how many worker-hours were required to distribute new telephone books to the first $70\\%$ of the households? $88.89$
(d) how many worker-hours were required to distribute new telephone books to the entire community? $133.33$
(e) what percentage of the households in the community had received new telephone books by the time $109$ worker-hours had been expended? $84$

Explanation:

Step1: Identify domain mathematically

The function $W(x)=\frac{800x}{700-x}$ is undefined when denominator is 0:
$700 - x = 0 \implies x=700$. Also, $x$ represents a percentage, so mathematically $x
eq 700$, but context restricts $x$ to percentage values. However, for the pure domain of the function (ignoring context first), we also can't have negative values that don't make sense for the input, but strictly mathematically, domain is all real numbers except $x=700$. But in the context of percentage, $x$ is a percentage, so we also consider that $x$ can't be negative (since percentage can't be negative) and can't exceed 100 for practicality, but the pure function domain (part a) is all real numbers where $700-x
eq 0$, but since $x$ is a percentage, the valid domain for the function in the context of the problem's input is $(0, 100)$ is incorrect. The correct mathematical domain considering $x$ is a percentage (so $x$ is between 0 and 100, and denominator not zero, which 700 is outside 0-100, so domain is $[0,100]$? No, wait, when $x=100$, denominator is $700-100=600
eq 0$, so function is defined at $x=100$. So the correct domain (pure function, part a) is all real numbers except $x=700$, but since $x$ is a percentage, the relevant domain is $[0, 100]$. But the given answer was marked wrong, so let's correct:

Step1: Fix part(a) domain

Function undefined at $700-x=0 \implies x=700$. Since $x$ is a percentage, $x \in [0,100]$, which does not include 700. So domain is $[0, 100]$.

Step2: Part(b) practical interpretation

$x$ is percentage of households, so $0 \leq x \leq 100$.

Step3: Part(c) Calculate $W(70)$

Substitute $x=70$ into $W(x)$:
$W(70)=\frac{800 \times 70}{700 - 70}=\frac{56000}{630} \approx 88.89$

Step4: Part(d) Calculate $W(100)$

Substitute $x=100$ into $W(x)$:
$W(100)=\frac{800 \times 100}{700 - 100}=\frac{80000}{600} \approx 133.33$

Step5: Part(e) Solve for $x$ when $W(x)=109$

Set $\frac{800x}{700 - x}=109$
Multiply both sides by $700-x$:
$800x = 109(700 - x)$
$800x = 76300 - 109x$
$800x + 109x = 76300$
$909x = 76300$
$x=\frac{76300}{909} \approx 84$

Answer:

(a) $[0, 100]$
(b) $0 \leq x \leq 100$
(c) $88.89$
(d) $133.33$
(e) $84$