Calculate the Rate of Change?

The amount of money a company has in the bank after t years is 180 +3t -t^2 thousand dollars.

(a) How much did the bank account increase during year 4? (Hint: Year 1 starts at t=0 and ends at t=1.)

(b) What was the average rate of increase of money in the account during year 4?

(thousands of dollars per year)

(c) What was the instantaneous rate of increase of money in the account at the start of each of the four quarters year 4?

(d) After how many years did the bank account run out of money?

Calculate the Rate of Change?

a) At the beginning of the fourth year, the account had 180 + 3(3) - (3)^2 thousand dollars in it - that%26#039;s 180 thousand dollars. At the end of the year it had 180 + 3(4) - (4)^2 thousand dollars in it - that%26#039;s 176. So the amount of money in the account increased by -4 thousand dollars that year (or decreased by 4 thousand dollars).

b) The amount went up by -4 thousand dollars during year 4. So unless I%26#039;m missing something the average rate of increase during year 4 was -4 thousand dollars per year.

c) You%26#039;d have to work that out using t=3.25, t=3.5, etc.

d) 180 + 3t - t^2 = 0, that%26#039;s a quadratic

rewrite it as t^2 - 3t - 180 = 0

factors of 180 that differ by 3: 15 and 12

(t + 12)(t - 15) = 0

t = 15 or -12 years

Negative years don%26#039;t make sense here, so the only possible answer is 15 years.

Hope that helps (and that it%26#039;s right!)