We want to use the given functions to create another function that models the revenue of the worker as a function of the time he works.
The solutions are:
A) r(x) = x^2 - 150
B) $1,850
C) The difference quotient is equal to 2*x.
We have two functions:
f(x) = 36*x^2 - 150
f(x) is the amount of money that he wins for decorating x rooms.
g(x) = (1/6)*x
g(x) is the number of rooms that he decorates in x hours.
So the revenue as a function of time can be given by evaluating f(x) in g(x).
A) we get:
r(x) f( g(x)) = 36*[(1/6)*x]^2 - 150 = x^2 - 150
r(x) = x^2 - 150
B) If he works for 45 hours, we just need to replace x by 45 in the revenue equation:
r(45) = 45^2 - 150 = 1,875
Meaning that he would win $1,875 for 45 hours of work.
C) the difference quotient for a function f(x) is given by:
[tex]\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]
For the case of r(x) we have:
[tex]\lim_{h \to 0} \frac{r(x + h) - r(x)}{h} = \lim_{h \to 0} \frac{(x + h)^2 - 150 - x^2 + 150}{h} = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = 2x[/tex]
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