Textbook Question
2–9. Integrals Evaluate the following integrals.
∫₁⁴ (10^{√x} / √x) dx
46
views
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooks
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.26b
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.RE.26bChapter 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 guidance1
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 \(t\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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.
Recommended video:
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.
Recommended video:
02:32Verifying 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.
Recommended video:
09:29Exponential Growth & Decay
Related Practice
Textbook Question
Radioactive decay The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?
106
views
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.
85
views
Textbook Question
2–9. Integrals Evaluate the following integrals.
∫ dx / √(x² − 9),x > 3
75
views
Textbook Question
2–9. Integrals Evaluate the following integrals.
∫₀¹ (x² / (9 − x⁶)) dx
58
views
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.
73
views