Science:Math Exam Resources/Courses/MATH105/April 2011/Question 05 (c)

MATH105 April 2011

  •  Q1 (a)  •  Q1 (b)  •  Q2  •  Q3  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q6  •  Q7  •  Q8 (a)  •  Q8 (b)  •  Q8 (c)  •  Q9 (a)  •  Q9 (b)  •  Q9 (c)  •  Q9 (d)  •  Q9 (e)  •  Q9 (f)  •  Q9 (g)  •  

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Question 05 (c)

You open a savings account under the Escalator Plan with an initial deposit of $\$P$. The advantage of this plan is that the interest rate is not fixed, but grows proportionally with time as long as the account is alive. In other words, the money in the account collects interest at the annual rate of $rt$ at time $t,$ compounded continuously (here $r$ is a constant). You also keep adding money to the account in the form of a continuous deposit of $\$S(t)$ at time $t.$ With the same conditions as in part (b), how long does it take for the account size to reach $\$100e^{2}$ after the initial deposit?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
How can you transform this question into an equation to be solved?

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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. We are looking for the time $t$ such that $A(t)=100e^{2}$. A computation shows ${\begin{aligned}A(t)&=100e^{2}\\100e^{0.005\cdot t^{2}}&=100e^{2}\\e^{0.005\cdot t^{2}}&=e^{2}\\0.005\cdot t^{2}&=2\\t^{2}&=2/0.005=400\\t&=\pm {\sqrt {400}}=\pm 20\end{aligned}}$ Since the question asks for the time $t$ after the initial deposit, the negative solution isn't feasible. We arrive at $t=20$ years.

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