University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 1 - Section 1.5 - Exponential Functions - Exercises - Page 38: 34

Answer

The investment would triple in value after around 20 years.

Work Step by Step

We have an investment with a $5.75\%$ interest compounded continuously. And we would suppose that this $5.75\%$ interest is an annual one. That means every year, that investment would increase by $5.75\%$ of the amount available. So, if we take the inital value of the investment to be $a_i$: - After first year, it would increase by $5.75\%$ of $a_i$ -> now its value is $105.75\%a_i$ or $1.0575a_i$ - After second year, it would increase by $5.75\%$ of $1.0575a_i$ -> its value now is $1.0575\times1.0575a_i=(1.0575)^2a_i$. Therefore, if we take $t$ to be the amount of time for the investment to triple the money (which equals to $3a_i$), we have this equation: $$(1.0575)^ta_i=3a_i$$ $$(1.0575)^t=3$$ Here, we use graphing calculator and find out that $$t=19.65\approx20(years)$$ So the investment would triple in value after around 20 years.
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