8. a printing plant that produces baseball cards has a monthly overhead of $6,000. it costs 18 cents to…

8. a printing plant that produces baseball cards has a monthly overhead of $6,000. it costs 18 cents to print each card, and the cards sell for 30 cents each. how many cards must the printing plant sell each month in order to make a profit? (a) 30,000 (b) 40,000 (c) 50,000 (d) 60,000

8. a printing plant that produces baseball cards has a monthly overhead of $6,000. it costs 18 cents to print each card, and the cards sell for 30 cents each. how many cards must the printing plant sell each month in order to make a profit? (a) 30,000 (b) 40,000 (c) 50,000 (d) 60,000

Answer

Explanation:

Step1: Set up profit - cost equation

Let $x$ be the number of cards sold. The cost function $C(x)=6000 + 0.18x$ (in dollars, since overhead is $6000$ and cost per card is $0.18$). The revenue function $R(x)=0.30x$ (since selling - price per card is $0.30$). To make a profit, $R(x)>C(x)$, so $0.30x>6000 + 0.18x$.

Step2: Solve the inequality

Subtract $0.18x$ from both sides: $0.30x-0.18x>6000$. This simplifies to $0.12x>6000$.

Step3: Isolate $x$

Divide both sides of the inequality $0.12x>6000$ by $0.12$: $x>\frac{6000}{0.12}$.

Step4: Calculate the value of $x$

$\frac{6000}{0.12}=50000$. So the plant must sell more than $50000$ cards to make a profit.

Answer:

D. 60,000