company is hoping to expand its facilities but needs capital

company is hoping to expand its facilities but needs capital to do so. In an effort to position itself for expansion in 3 years, the company will direct half of its profits into investments in a continuous manner. The company\'s profits for the past 5 years are shown in the table. The company\'s current yearly profit is $1,130,000. (The currrent year corresponds to t = 0.)

Profit over the Past Five Years

Consider the following profit scenarios and answer the questions.

(a) The profit for the next 3 years follows the trend shown in the table.

(i) Write the quadratic function that describes the flow of the company\'s investments. (Align the input such that 5 years ago corresponds to t = 5. Round all numerical values to three decimal places.)
R(t) = thousand dollars

(ii) Calculate the capital the company will have saved after 3 years of investing at 6.4% annual interest compounded continuously. (Round your answer to three decimal places.)
$ 2 thousand

(b) The profit increases each year for the next 3 years by the same percentage that it increased in the current year.

(i) Write the function 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 3 years of investing at 6.4% annual interest compounded continuously. (Round your answer to three decimal places.)
$ 4 thousand

(c) The profit remains constant at the current year\'s level.

(i) Write the function that describes the flow of the company\'s investments.
R(t) = thousand dollars

(ii) Calculate the capital the company will have saved after 3 years of investing at 6.4% annual interest compounded continuously. (Round your answer to three decimal places.)
$ 6 thousand

(d) The profit increases each year for the next 3 years by the same fixed amount that it increased this year.

(i) Write the function that describes the flow of the company\'s investments.
R(t) = thousand dollars

(ii) Calculate the capital the company will have saved after 3 years of investing at 6.4% annual interest compounded continuously. (Round your answer to three decimal places.)
$ 8 thousand

Years Ago	5	4	3	2	1
Profit (thousand dollars)	870	890	940	980	1060

Solution

In an effort to position itself for expansion in 3 years, the company will direct half of its profits into investments in a continuous manner.

that is , half of the profit over the past five years will go into the investment for the expansion

a>

(i)

the quadratic function could be defined as R(t) = at^2 + bt + c

where a , b and c are constants that need to be found

t is the time in years , and R(t) is the flow of the companies investment that depends on t

now when t = -5 (five years ago from today) , R(t) = 870/2 = 435

=> R(t) = at^2 + bt + c

435 = 25a - 5b + c   -------------> (1)

when t = -4 , R(t) = 890/2 = 445

=> 445 = 16a - 4b + c ----------> (2)

when t = -2 , R(t) = 980/2 =495

=> 490 = 4a - 2b + c -----------> (3)

now solving equations (1) , (2) , (3) for a , b and c

=> a = 25/6 , b = 95/2 and c = 1705/3

=> R(t) = 25/6*t^2 + 95/2*t + 1705/3

or R(t) = 4.167t^2 + 47.5t + 568.333

(ii)

since the annual interest is compounded continuously so will use the formula

A = Pe^(r*t)

P = investment = $ 2000

A is the amount after time t in years

r = 6.4% = .064

t = 3 years

=> A = 2000*e^(.064*3) = 2423.341

=> the company would have saved = $ 2423.341

b> The profit increases each year for the next 3 years by the same percentage that it increased in the current year

i>

=> investment function will follow a linear model of the form R(t)= mt + b

t = time in years and m and b are constants

=> at t =0 , R(t) = 1130/2 = 565

=> 565 = 0+b

=> R(t) = m*t + 565

here m is the constant rate with which the profit is being invested

ii> A = Pe^(rt)

P = $ 4000

t = 3 years

r = 6.4 % per annum = .064

=> A = 4000*e^(.064*3) = 4846.682

=> the company would have saved $ 4846.682