Five thousand dollars is deposited into a savings account at 7 5% interest compo

Question

Five thousand dollars is deposited into a savings account at 7 5% interest compounded continuously. What is the formula for A(t), the balance after t years? What differential equation is satisfied by A(t), the balance after t years? How much money will be in the account after 3 years? When will the balance reach $7000? How fast is the balance growing when it reaches $7000? A(t) = squarebox A'(t) = squarebox $ squarebox (Round to the nearest cent as needed.) After squarebox years the balance will reach $7000. (Round to one decimal place as needed.) The investment is growing at the rate of $ squarebox per year. (Type an integer or decimal rounded to two decimal places as needed.)

Explanation / Answer

Formula for continuous compounding is given by
A= P ert ; Where the terms have the usual meanings as in simple or compound interest;

i) A (t) = 5000 et*0.075
ii) A'(t) = 5000*0.075 * et*0.075 = 375 et*0.075
iii) After 3 years, A(3) = 5000 e3*0.075 = 5000 e0.225 = 5000*1.2523=6261.5 $
iv) For A = 7000; 7000= 5000 et*0.075; Thus t*0.075 = ln (7/5) = 0.3364; Thus t= 0.3364/0.075=4.48 years;
Thus amount will reach 7000 after 4.48 years;
v) At t=4.48 years; A'(t) = 375 e4.48 * 0.075 = 375 e0.336 = 375*1.399 = 524.75$;
Thus when A = 7000 it is growing at the rate of 524.75 $ per year

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