Respuesta :
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.
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
