Muralist Cost Function: $360 for First 3 Hours

Hello! I am Acemy. Based on the image provided, your problem asks to identify the linear function that models the total cost of hiring a muralist, given a base price for the first three hours and an additional hourly rate.

Answer

The correct option is A. The function models the total cost by adding the fixed cost for the initial hours to the variable rate applied specifically to the number of additional hours worked.

Explanation

1. Identify the variables and constraints: We are given that $x$ represents the total hours, where $x \ge 3$. The cost $f(x)$ consists of a fixed fee for the first 3 hours and a variable fee for the remaining hours $(x - 3)$. $f(x) = 360 + m(x - 3)$ This formula represents the fixed base cost plus the hourly rate $m$ multiplied by the number of hours exceeding the initial three.

2. Calculate the hourly rate ($m$): We know that at $x = 5$, the total cost $f(5) = 540$. We substitute these values into our equation to solve for the hourly rate $m$: $540 = 360 + m(5 - 3)$ This equation scales the initial cost by the remaining hours to solve for the unknown slope. $540 = 360 + 2m$ $180 = 2m$ $m = 90$ This identifies that the muralist charges $90 per hour for every hour after the first three. ⚠️ This step is required on exams to define the rate of change correctly.

3. Simplify the function: Now, substitute $m = 90$ back into the general equation: $f(x) = 360 + 90(x - 3)$ This incorporates the identified hourly rate into the cost model. $f(x) = 360 + 90x - 270$ $f(x) = 90x + 90$ This result is the simplified slope-intercept form used to find the total cost for any $x \ge 3$.

Option	Function	Correctness
A	$f(x) = 90x + 90$	Correct
B	$f(x) = 90x + 360$	Incorrect (fails $f(5)=540$)
C	$f(x) = 108x + 90$	Incorrect (wrong rate)
D	$f(x) = 108x + 360$	Incorrect (wrong rate)

Final Answer

$\boxed{f(x) = 90x + 90}$

Common Mistakes

• Ignoring the "additional hours" constraint: Many students mistakenly use $f(x) = m(x) + b$ without accounting for the fact that the hourly fee only applies to hours beyond the first three, leading them to calculate an incorrect intercept.
• Calculation Error in Rate: Failing to subtract the first 3 hours from the total time (5 hours) results in dividing by 5 instead of 2, leading to an incorrect slope. Always ensure your variable portion represents only the time that incurs the additional cost.