An online retail company determined that their cost for each day, x, can be determined by the function C(x) = 2x2 – 20x + 80 and

Question

10 months ago

5

their revenue by the function R(x) = –x2 + 5x + 58. How many days will it take for the company to earn a profit?

1 answer:

sammy [17] · Answer 1 · 2024-04-05T04:10:01+03:00

The number of days the company earns a profit is the second day till the 7th day

<h3>How to determine the number of days the company earns a profit?</h3>

From the question, we have the following functions that can be used in our computation:

Cost function, C(x) = 2x2 – 20x + 80

Revenue function, R(x) = –x2 + 5x + 58

These functions can be properly represented as

Cost function, C(x) = 2x² – 20x + 80

Revenue function, R(x) = –x² + 5x + 58

When the company makes a profit, the revenue of the company exceeds the cost

This means that

R(x) > C(x)

Substitute the known values in the above equation, so, we have the following representation

–x² + 5x + 58 > 2x² – 20x + 80

Evaluate the like terms

–3x² + 25x - 22 > 0

Expand

–3x² + 3x + 22x - 22 > 0

Factorize the inequality

–3x(x - 1) + 22(x - 1) > 0

So, we have

(-3x + 22)(x - 1) > 0

This gives

-3x + 22 > 0 and x - 1 > 0

Evaluate

-3x > -22 and x > 1

Divide

x < 7.3 and x > 1

This can be combined as

1 < x ≤ 7

Hence, the company makes profit from the second day till the 7th day