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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.RE.26b
7장, 문제 7.RE.26b

Savings account A savings account advertises an annual percentage yield (APY) of 5.4%, which means that the balance in the account increases at an annual growth rate of 5.4%/yr.


b. What is the doubling time of the balance?

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Recognize that the problem involves exponential growth, where the balance grows continuously at an annual rate of 5.4%. The doubling time is the time it takes for the balance to become twice its initial amount.
Use the formula for exponential growth: \(A = A_0 e^{rt}\), where \(A\) is the amount after time \(t\), \(A_0\) is the initial amount, \(r\) is the growth rate (expressed as a decimal), and \(t\) is time in years.
Set \(A = 2 A_0\) to represent doubling, so the equation becomes \(2 A_0 = A_0 e^{rt}\). Simplify this to \(2 = e^{rt}\).
Take the natural logarithm of both sides to solve for \(t\): \(\ln(2) = rt\). Then, solve for \(t\) by dividing both sides by \(r\): \(t = \frac{\ln(2)}{r}\).
Substitute the given growth rate \(r = 0.054\) (since 5.4% = 0.054) into the formula to find the doubling time \(t\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Exponential Growth

Exponential growth describes a process where a quantity increases by a fixed percentage over equal time intervals. In the context of savings accounts, the balance grows by a constant rate annually, leading to compounding effects that cause the amount to increase faster over time.
추천 영상:
09:29
Exponential Growth & Decay

Doubling Time

Doubling time is the period required for an initial amount to grow to twice its size at a constant growth rate. It can be calculated using the formula involving logarithms or approximated by the Rule of 72, which divides 72 by the annual growth rate percentage.
추천 영상:
02:32
Verifying Solutions of Differential Equations Example 3

Logarithms in Growth Calculations

Logarithms are used to solve for time in exponential growth equations because they allow us to isolate the variable in the exponent. When finding doubling time, logarithms help convert the exponential equation into a linear form to solve for the unknown time.
추천 영상:
09:29
Exponential Growth & Decay
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