Question

The profits of mr cash's company is represented by the equation -3(t-3)^2+23, where 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?

Answer 1

(a)3 years

(b)22

(c)5.77 years

Explanation:

The profits of Mr Cash's company is represented by the equation [TeX] -3(t-3)^2+23[/TeX].

(a)We are to determine in what year the company will make maximum profit.

Simplifying:

[TeX] -3(t-3)^2+23=-3t^2+18t-4[/TeX].

The maximum profit will occur at the point where the derivative

of the profit function is 0.

If [TeX] P=-3t^2+18t-4[/TeX]

The derivative:

P'=-6t+18

-6t+18=0

-6t=-18

t=3 years

In the third year, the business will show maximum profit.

(b)At t=3, we want to derive the maximum profit.

[TeX] P=-3t^2+18t-4[/TeX]

P=-3(3²)+18(3)-5

=-27+54-5

P=22

Maximum Profit=22 (in the appropriate unit)

(c)When the business yields no profit, it is too late to sell the business. This occurs at the point where the profit function equals zero.

[TeX] P=-3t^2+18t-4=0[/TeX]

Solving the resulting quadratic equation using a calculator,

t=5.77 or 0.23

The first value is lower than the time at which he makes maximum profit, we ignore that point.

Therefore in 5.77 years, it would be too late to sell his business.