Question

a marketing team is going to buy promotional coffee mugs as giveaways. the manufacturing company charges a fixed $400 setup fee and $7 per mug. complete an equation for the average cost per coffee mug if x mugs are made.
average cost =
how many coffee mugs should be produced so that the average cost per coffee mug is less than $31?

Explanation:

Step1: Find the average cost formula

The total cost \(C\) is the sum of the fixed cost and the variable cost. The fixed cost is \(400\) dollars and the variable cost is \(7x\) dollars (since it's \(7\) dollars per mug and \(x\) mugs are made). So, \(C = 400+7x\). The average cost \(A(x)\) is the total cost divided by the number of mugs, so \(A(x)=\frac{400 + 7x}{x}=\frac{400}{x}+7\).

Step2: Set up the inequality

We want the average cost per coffee mug to be less than \(31\) dollars. So, we set up the inequality \(\frac{400}{x}+7<31\).

Step3: Solve the inequality

Subtract \(7\) from both sides: \(\frac{400}{x}<31 - 7\), which simplifies to \(\frac{400}{x}<24\).
Since \(x>0\) (number of mugs can't be negative), we can multiply both sides by \(x\) without changing the inequality sign: \(400<24x\).
Then divide both sides by \(24\): \(x>\frac{400}{24}=\frac{50}{3}\approx16.67\).
Since \(x\) represents the number of mugs and it must be a whole number, we take \(x = 17\) (because if \(x = 16\), \(\frac{400}{16}+7=25 + 7=32>31\)).

Answer:

\(17\) mugs