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?

Question

25-01-2020
Mathematics

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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?

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ChiKesselman ChiKesselman · Answer 1 · 2020-01-27T08:16:15+01:00

Answer:

a) Domain = [tex](-\infty,400)\cup (400,\infty)[/tex]

b) [tex]x \in [0,100][/tex]

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:

[tex]W(x) = \dfrac{250x}{(400-x)}[/tex]

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 = [tex](-\infty,400)\cup (400,\infty)[/tex]

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

[tex]x \in [0,100][/tex]

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

[tex]W(70) = \dfrac{250(70)}{(400-70)} = 53.03[/tex]

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

[tex]W(100) = \dfrac{250(100)}{(400-100)} = 83.33[/tex]

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

[tex]3 = \dfrac{250x}{(400-x)}\\\\1200-3x = 250x\\253x = 1200\\\\x = \dfrac{1200}{253} = 4.74\%[/tex]

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

xero099 xero099 · Answer 2 · 2022-01-10T22:35:28+01:00

a) The domain of the function [tex]W[/tex] is [tex]\mathbb{R}-\{400\}[/tex].

b) The subset of the domain that have a practical interpretation is [tex][0, 100][/tex].

c) 53.030 worker-hours are required to distribute new telephone books to the first 70 % of the households.

d) 83.333 worker-hours are required to distribute new telephone books to 100 % of the households.

e) 4.743 % have received the new telephone books by the time 3 worker-hours had been expended.

Procedure - Functional analysis on an expression for the number of worker-hours to distribute new telephone books in a rural community

POINT A

Determination of the domain of the function

Mathematically speaking, the domain of the function is the set of values of [tex]x[/tex] such that a value of [tex]W[/tex] exists. In the case of rational functions, the domain is all real numbers except for all values of [tex]x[/tex] such that the denominator becomes zero.

The value of [tex]x[/tex] such that rational function become undefined is:

[tex]400 - x = 0[/tex]

[tex]x = 400[/tex]

Hence, the domain of the function [tex]W[/tex] is [tex]\mathbb{R}-\{400\}[/tex]. [tex]\blacksquare[/tex]

Point B

Determination of context-based domain

Percentages are positive real numbers between 0 and 100. Hence, the subset of the domain that have a practical interpretation is [tex][0, 100][/tex]. [tex]\blacksquare[/tex]

Point C

Determination of required worker-hours (I)

If we know that [tex]W(x) = \frac{250\cdot x}{400-x}[/tex] and [tex]x = 70[/tex], then we have that the required worker-hours are:

[tex]W(70) = \frac{250\cdot (70)}{400-70}[/tex]

[tex]W(70) = 53.030[/tex]

53.030 worker-hours are required to distribute new telephone books to the first 70 % of the households. [tex]\blacksquare[/tex]

Point D

Determination of required worker-hours (II)

If we know that [tex]W(x) = \frac{250\cdot x}{400-x}[/tex] and [tex]x = 100[/tex], then we have that the required worker-hours are:

[tex]W(100) = \frac{250\cdot (100)}{400-100}[/tex]

[tex]W(100) = 83.333[/tex]

83.333 worker-hours are required to distribute new telephone books to 100 % of the households. [tex]\blacksquare[/tex]

Point E

Determination of percentage of the household that receive the telephone books

If we know that [tex]W(x) = \frac{250\cdot x}{400-x}[/tex] and [tex]W(x) = 3[/tex], then we have that the percentage of the households that received the new telephone books are:

[tex]3 = \frac{250\cdot x}{400-x}[/tex]

[tex]1200-3\cdot x = 250\cdot x[/tex]

[tex]253\cdot x = 1200[/tex]

[tex]x = 4.743[/tex]

4.743 % have received the new telephone books by the time 3 worker-hours had been expended. [tex]\blacksquare[/tex]

To learn more on rational functions, we kindly invite to check this verified question: https://brainly.com/question/15324782