Skip to main content
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.20a
Chapter 7, Problem 7.RE.20a

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.

Verified step by step guidance
1
Identify the exponential growth model formula: \(P(t) = P_0 \cdot e^{rt}\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(r\) is the growth rate, and \(t\) is the time in years.
Assign the known values: initial population \(P_0 = 1.3\) million, population after 10 years \(P(10) = 1.45\) million, and \(t = 10\) years.
Use the known values to set up the equation \(1.45 = 1.3 \cdot e^{10r}\) and solve for the growth rate \(r\) by isolating \(r\).
Once \(r\) is found, use the exponential model again to estimate the population after 20 years by calculating \(P(20) = 1.3 \cdot e^{20r}\).
Interpret the result as the estimated population in millions after 20 years, based on the constant annual growth rate.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

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

Exponential Growth Model

Exponential growth describes a process where the quantity increases at a rate proportional to its current value, modeled by P(t) = P_0 * e^(rt). Here, P_0 is the initial population, r is the growth rate, and t is time. This model is essential for predicting future population sizes when growth is continuous and constant.
Recommended video:
09:29
Exponential Growth & Decay

Determining the Growth Rate

To use the exponential model, the growth rate r must be found from known data points. Given initial and future populations at specific times, r can be calculated by rearranging the formula: r = (1/t) * ln(P(t)/P_0). This step is crucial to accurately estimate future populations.
Recommended video:
04:16
Intro To Related Rates

Using the Model for Prediction

Once the growth rate is known, the exponential model can predict population at any future time by substituting t into P(t) = P_0 * e^(rt). This allows estimation of the population after 20 years, assuming the growth rate remains constant over time.
Recommended video:
08:14
Using The Acceleration Function
Related Practice
Textbook Question

2–9. Integrals Evaluate the following integrals.


∫₁⁴ (10^{√x} / √x) dx

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?

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
views
Textbook Question

2–9. Integrals Evaluate the following integrals.


∫ dx / √(x² − 9),x > 3

1
views
Textbook Question

2–9. Integrals Evaluate the following integrals.


∫₀¹ (x² / (9 − x⁶)) dx

Textbook Question

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?

1
views