The profits of Mr. Cash's company is represented by the equation 푃 푡 =−3푡 ) +18푡−4, where 푃(푡) is the amount of profit in hundreds of thousands of dollars and 푡 is the number of years of operation. He realizes his company is on the down turn and wishes to sell before he ends up in debt.

- Mr Cash's company business shows the maximum profit in 3 years after starting the business.

- The maximum profit = 23 thousand dollars = $23,000

- Anytime beyond 5.77 years will be too late to sell the business as the profit margins have entered negative values.

Step-by-step explanation:

Complete Correct Question

The profits of Mr. Cash’s company is represented by the equation

p(t)=-3t^2+18t-4,

where p(t) is the amount of profit in hundreds of thousands of dollars and t is the number of years of operation. He realizes his company is on the down turn and wishes to sell before he ends up in debt.

In what year of operation does Mr. Cash’s business show maximum profit? What is the maximum profit? What time will it be too late to sell business?

Solution

The profit function is given as

p(t) = -3t² + 18t - 4

p(t) = profit in thousand of dollars

t = time in years.

To find the year where the maximum profit is obtained, we use differentiation analysis to obtain the maximum of the profit function and the corresponding time where it occurs.

At the maximum of the profit function,

(dp/dt) = 0 and (d²p/dt²) < 0

p(t) = -3t² + 18t - 4

(dp/dt) = -6t + 18

At maximum profit

(dp/dt) = -6t + 18 = 0

6t = 18

t = (18/6) = 3 years.

(d²p/dt²) = -6 < 0

Hence, this value of t corresponds to the maximum value of the profit function.

The maximum profit is then obtained when t = 3

p(t) = -3t² + 18t - 4

p(3) = -3(3²) + 18(3) - 4 = 23 thousand dollars

The business will be late to sell when the profits from the business start to run into negative values. So, we obtain the time when the profit margins reaches 0.

p(t) = -3t² + 18t - 4 = 0

-3t² + 18t - 4 = 0

Solving the quadratic equation

t = 0.23 years or t = 5.77 years

Since the profits first start rising from the value at t=0 up till the maximum profit at t=3, 0.23 years isn't a realistic point to sell the business as this is part of the onset of profits, but beyond 5.77 years on the other hand represents when the business stops to bringing in profits and leads to losses.

Hope this Helps!!!

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