Question 662·Easy·Algebra

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A printing company uses the function

c(x)=12x+60

to determine the total cost $c(x)$ , in dollars, of producing $x$ custom coffee mugs. If the total cost is $300, how many mugs are produced?

In a linear cost function, the input counts items and the output is the total cost. When the total is given but the item count is asked for, set the rule equal to that output and solve for the input. A Desmos regression can find that input. Dividing the entire total by the per-item charge counts too many items because it ignores the fixed fee.

Hints

Hint 1
The output of $c(x)$ is the dollar total, while $x$ is the mug count. The dollar amount is given, so what must $c(x)$ equal?
Hint 2
The $60$ is a fixed cost, paid once. Only the per-mug charge gets multiplied by $x$ . After writing the total-cost equation, solve for the mug count.

Step-by-step

Set the cost rule equal to the total
Step 1
Identify the given output
The output $c(x)$ is the cost in dollars, and the input $x$ counts mugs. The stated total is $300, so $c(x)=300$ . Putting $300$ in for $x$ instead would mean producing 300 mugs.
Step 2
Write the total-cost equation
Replace $c(x)$ with the given rule:
$12x+60=300$
The $60$ is a fixed cost, paid once. The $12x$ charges $12 for each mug, so the $60$ does not get multiplied by $x$ .
Step 3
Find the mug count in Desmos
Type $12x_{1}+60\sim300$ in Desmos. The $x_{1}$ names the unknown input, and $\sim$ starts a regression that finds the value making both sides match. Desmos reports $x_{1}=20$ under PARAMETERS, so the company produces 20 mugs. Choice D.

Question 662·Easy·Algebra

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