Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 5 - Exponential and Logarithmic Functions - 5.7 Financial Models - 5.7 Assess Your Understanding - Page 321: 32



Work Step by Step

According to the Compound Interest Formula, where $P$ is the principal, the amount deposited, $r$ is the annual interest rate, $n$ is the number of times the interest is compounded annually, $t$ is the number of years, $A$ is the amount the loaner gets back after $t$ years: $A=P\cdot(1+\frac{r}{n})^{n\cdot t}.$ The investment is compounded annualy, hence $n=1$. Thus, the formula above becomes: $A=P\cdot(1+\frac{r}{1})^{1\cdot t}\\ A=P\cdot(1+r)^{t}.$ The given situation has $t=6$ years $A=2P$ because the investment doubles after $6$ years. Using the formula above gives: $2\cdot P=P\cdot(1+r)^6\\2=(1+r)^6.\\\sqrt[6] {2}=\sqrt[6] {(1+r)^6}\\\sqrt[6] 2=1+r\\r=\sqrt[6] 2-1$. Use a calculator to obtain: $r=\sqrt[6] 2-1\\r=1.122462-1\\r=0.122462\\r\approx12.25\%.$
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