A company is hoping to expand its facilities but needs capitalto do so. In an effort to position itself for expansion in 3 years,the company will direct half of its profits into investments in acontinuous manner. The company's profits for the past 5 years areshown in the table. The company's current yearly profit is$1,160,000. (The current year corresponds to t = 0.)
Profit over the Past Five Years
| Years Ago | 5 | 4 | 3 | 2 | 1 |
|---|
Profit
(thousand dollars) | 860 | 900 | 930 | 1000 | 1060 |
|---|
Consider the following profit scenarios and answer thequestions.
(a) The profit for the next 3 years follows the trend shown inthe table.
(i) Write the quadratic function that describes the flow of thecompany's investments. (Align the input such that 5 years agocorresponds to t = ?5. Round all numerical values to threedecimal places.)
R(t) = ______thousand dollars
(ii) Calculate the capital the company will have saved after 3years of investing at 6.4% annual interest compounded continuously.(Round your answer to three decimal places.)
$ _____thousand
(b) The profit increases each year for the next 3 years by the samepercentage that it increased in the current year.(i) Write thefunction that describes the flow of the company's investments.(Round all numerical values to three decimal places.)
R(t) = ______ thousand dollars
(ii) Calculate the capital the company will have saved after 3years of investing at 6.4% annual interest compounded continuously.(Round your answer to three decimal places.)
$ ______ thousand
(c) The profit remains constant at the current year's level.(i)Write the function that describes the flow of the company'sinvestments.
R(t) = ______thousand dollars
(ii) Calculate the capital the company will have saved after 3years of investing at 6.4% annual interest compounded continuously.(Round your answer to three decimal places.)
$ _____ thousand
(d) The profit increases each year for the next 3 years by the samefixed amount that it increased this year.(i) Write the functionthat describes the flow of the company's investments.
R(t) = ______thousand dollars
(ii) Calculate the capital the company will have saved after 3years of investing at 6.4% annual interest compounded continuously.(Round your answer to three decimal places.)
$_______ thousand
