Question

An interior designer wants to decorate a newly constructed house. The function f(x) = 36×2 – 150 represents the amount of money he earns per room decorated, where x represents the number of rooms he designs. The function g(x)=1/6x represents the number of rooms the interior designer decorates, where x is the number of hours he works.
: Determine the amount of money the interior designer will make decorating the house as a function of hours he works.
: If the newly constructed house requires 45 hours of work, how much will the interior designer earn? Show all necessary calculations.
Part C: Determine an expression to represent the difference quotient for the function found in . Show all necessary work.

Answer 1

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:

$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

For the case of r(x) we have:

$\lim_{h \to 0} \frac{r(x + h) - r(x)}{h} = \lim_{h \to 0} \frac{(x + h)^2 - 150 - x^2 + 150}{h} = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = 2x$

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