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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.26b
Chapter 7, Problem 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?

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.


a. Use an exponential model to estimate the population in 20 years. Assume the annual growth rate is constant.

Textbook Question

2–9. Integrals Evaluate the following integrals.


∫₁⁴ (10^{√x} / √x) dx

Textbook Question

Moore’s Law In 1965, Gordon Moore observed that the number of transistors that could be placed on an integrated circuit was approximately doubling each year, and he predicted that this trend would continue for another decade. In 1975, Moore revised the doubling time to every two years, and this prediction became known as Moore’s Law.


a. In 1979, Intel introduced the Intel 8088 processor; each of its integrated circuits contained 29,000 transistors. Use Moore’s revised doubling time to find a function y(t) that approximates the number of transistors on an integrated circuit t years after 1979.

1
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Textbook Question

2–9. Integrals Evaluate the following integrals.


∫ dx / √(x² − 9),x > 3

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Textbook Question

Caffeine An adult consumes an espresso containing 75 mg of caffeine. If the caffeine has a half-life of 5.5 hours, when will the amount of caffeine in her bloodstream equal 30 mg?

Textbook Question

Population growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now.


b. Find the doubling time of the population.