Question

George has opened a new store and he is monitoring its success closely. He has found that this store’s revenue each month can be modeled by r(x)=x2+5x+14 where x represents the number of months since the store opens the doors and r(x) is measured in hundreds of dollars. He has also found that his expenses each month can be modeled by c(x)=x2−4x+5 where x represents the number of months the store has been open and c(x) is measured in hundreds of dollars. What does (r−c)(5) mean about George's new store?

Answer 1

• (r - c)(5) is the amount of profit generated by the George's store 5 months after it has been open.

• The total amount of profit 5 months after the store has been open is $5,400.

Explanation:

Given functions:

`r(x)=x^2+5x+14`

`c(x)=x^2-4x+5`

where:

• r(x) is the store's revenue each month (in hundreds of dollars).
• c(x) is the store's expenses each month (in hundreds of dollars).
• x is the number of months since the store has been open.

Profit is calculated by subtracting expenses from revenue.

Therefore, the composite function for profit is:

• r(x) - c(x) = (r - c)(x)

Since x represents the number of months the store has been open:
(r - c)(5) is the amount of profit generated by the George's store 5 months after it has been open.

To calculate the total amount of profit in the first 5 months, substitute x = 5 into the composite function:

`\begin{aligned}\implies (r-c)(5)&=r(5)-c(5)\\&=(5^2+5(5)+14)-(5^2-4(5)+5)\\&=(25+25+14)-(25-20+5)\\&=(50+14)-(5+5)\\&=64-10\\&=54\end{aligned}`

Since the revenue and expenses functions are measured in hundred of dollars, so too is the profit function. Therefore, the amount of profit generated in 5 months is $5,400.