A substance is initially 1000 grams and 20 days later it is

A substance is initially 1000 grams and 20 days later it is 2500 grams.
(a) Find a formula for the amount after t days if the substance grows linearly.

(b) Find a formula for the amount after t days if the substance grows exponentially.

(c) If the substance grows exponentially, by what percent does it increase perday?

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(5) Suppose that you have $40000 to deposit into an account. What percent interest rate would be needed so that the account would contain $1000000 after 30 years? Assume continuous compounding.

Solution

(5) The formula for continuous compounding of interest is A = Pe^rt , where P is the principal amount (initial investment), r is the rate of interest in decimals, t is the number of years and A is the amount after t years. Here, P = $ 40000, A = $ 1000000 and t = 30. Then, we have 1000000 = 40000e^30r where r is the rate of interest in decimal. Therefore, e^30r = 1000000/40000 = 250. On taking natural logarithms of both the sides, we have 30r ln e = ln 250 ( as lne^30r = 30rln e), or 30r = 5.521460918 ( as ln e = 1) so that r = 5.521460918 /30 = 0.184048697 or 18.4048697 % = 18.40 % (on rounding off to 2 decimal places)