Science:Math Exam Resources/Courses/MATH105/April 2010/Question 01 (k)

MATH105 April 2010

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[hide] Question 01 (k)
Suppose that money is deposited steadily into a savings account at the rate of $3000 per year. Determine the balance at the end of 5 years if the account pays 6% interest compounded continuously.

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[show] Hint
Can you describe the instantaneous rate of change of capital as a differential equation?

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[show] Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The instantaneous rate of change of capital has two components: We continuously add money at a rate of $3,000 per year; We earn 6% interest compounded continuously. If we denote by A(t) the balance of the account at time t, then we can write the differential equation ${\begin{aligned}{\frac {dA}{dt}}&=3000+6\%\,A(t)\\&=3000+0.06\,A(t)\end{aligned}}$ We solve this equation using separation of variables: $\int {\frac {dA}{3000+0.06A}}=\int \,dt$ Solving both sides separately yields ${\frac {1}{0.06}}\ln \|3000+0.06A\|+C_{1}=t+C_{2}$ Since A is positive, as the amount of money in the account, we can drop the absolute value sign. This yields $\displaystyle \ln(3000+0.06A)=0.06t+C_{3}$ or, equivalently $3000+0.06A=e^{0.06t+C_{3}}=e^{0.06t}e^{C_{3}}=C_{4}e^{0.06t}.$ C₄ is another arbitrary, but positive constant. Hence we obtain for A(t): $A(t)={\frac {1}{0.06}}\left(C_{4}e^{0.06t}-3000\right).$ Now it's time to think about the initial condition. Since the account does not start with any balance initially, we know that A(0) = 0. This yields for C₄ that $0={\frac {1}{0.06}}\left(C_{4}e^{0}-3000\right),\quad {\text{which implies}}\quad C_{4}=3000.$ With this we finally find the expression for the balance in dollars at t years: $A(t)={\frac {3000}{0.06}}\left(e^{0.06t}-1\right).$ This implies that at t = 5 years the balance is $A(5)={\frac {3000}{0.06}}\left(e^{0.06\cdot 5}-1\right)=17492.94$ Reality check: Without interest we would expect to see 5 x 3000 = 15 000 dollars in the account. The amount with interest should be in the same order of magnitude and a little larger.