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)=250x/(400−x). (a) What is the domain of the function W? (Give the domain in interval notation. If the answer includes more than one interval write the intervals separated by the "union" symbol, U.) (b) For what values of x does W(x) have a practical interpretation in this context? (c) How many worker-hours were required to distribute new telephone books to the first 70% of the households? (d) How many worker-hours were required to distribute new telephone books to the entire community? (e) What percentage of the households in the community had received new telephone books by the time 3 worker-hours had been expended?

Answer 1

Answer:

a) Domain = $(-\infty,400)\cup (400,\infty)$

b) $x \in [0,100]$

c) 53.03 worker hours

d) 83.33 worker hours

e) 83.33 worker hours

Step-by-step explanation:

We are given the following in the question:

$W(x) = \dfrac{250x}{(400-x)}$

where W(x) is the number of worker-hours required to distribute new telephone books to x% of the households in a certain rural community.

a) Domain of function.

The domain is the all the possible values of x that the function can take.

Domain = $(-\infty,400)\cup (400,\infty)$

b) Values of x

Since x is a percentage in reference to context, it can only take value upto 100. Also it cannot take any negative value.

So domain n reference to context will be

$x \in [0,100]$

c) worker-hours were required to distribute new telephone books to the first 70% of the households

$W(70) = \dfrac{250(70)}{(400-70)} = 53.03$

53.03 worker hours were required to distribute new telephone books to the first 70% of the households.

d) Worker hour for entire community

For entire community, x = 100

$W(100) = \dfrac{250(100)}{(400-100)} = 83.33$

83.33 worker hours were required to distribute new telephone books to the entire households.

e) Percentage of the households in the community for 3 worker hours

$3 = \dfrac{250x}{(400-x)}\\\\1200-3x = 250x\\253x = 1200\\\\x = \dfrac{1200}{253} = 4.74\%$

Thus, 4.74% of the households in the community had received new telephone books by the time 3 worker-hours had been expended.