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

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.2.46b

Chapter 7, Problem 7.2.46b

Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.
b. Suppose City C has a current population of y₀ < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y₀ and p such that Cities A and C have the same population in 10 years?

Verified step by step guidance

Express the population of City A after 10 years using the exponential growth formula: \(P_A(10) = 500,000 \times (1 + 0.03)^{10}\), where 0.03 represents the 3% growth rate.

Express the population of City C after 10 years similarly: \(P_C(10) = y_0 \times (1 + p)^{10}\), where \(y_0\) is the initial population of City C and \(p\) is its growth rate (expressed as a decimal).

Set the populations equal to find when City A and City C have the same population after 10 years: \(500,000 \times (1 + 0.03)^{10} = y_0 \times (1 + p)^{10}\).

Isolate \(y_0\) to express it in terms of \(p\): \(y_0 = \frac{500,000 \times (1 + 0.03)^{10}}{(1 + p)^{10}}\).

Interpret this relationship: for City C to catch up with City A in 10 years, its initial population \(y_0\) and growth rate \(p\) must satisfy the equation above, with \(y_0 < 500,000\) and \(p > 0.03\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Exponential Growth Model

Population growth at a constant percentage rate is modeled by exponential functions, where the population at time t is given by P(t) = P₀ * (1 + r)^t. Here, P₀ is the initial population, r is the growth rate per time period, and t is the number of time periods. This model helps predict future populations based on current data.

Equating Populations to Find Conditions

To find when two populations are equal, set their exponential growth expressions equal and solve for the unknown variable. This involves forming an equation like P₀₁*(1 + r₁)^t = P₀₂*(1 + r₂)^t and manipulating it to express one variable in terms of others, which reveals the relationship between initial populations and growth rates.

Logarithmic Functions for Solving Exponential Equations

When solving equations involving variables in exponents, logarithms are used to isolate the variable. Taking the natural log or log base 10 of both sides allows the exponent to be brought down as a multiplier, enabling algebraic manipulation to find unknown growth rates or initial populations.

Recommended video:

5:02

Solving Logarithmic Equations

Related Practice

Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.

b. Find the total energy (in MW-yr) used by the city over four full years beginning at t=0.

Textbook Question

Theorem 7.8

Differentiate sinh⁻¹ x = ln (x + √(x² + 1)) to show that d/dx (sinh⁻¹ x) = 1 / √(x² + 1).

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.

b. ln 0 = 1

Textbook Question

61–62. Points of intersection and area

b. Compute the area of the region described.

f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and x = ln 3

views

Textbook Question

A running model A model for the startup of a runner in a short race results in the velocity function v(t) = a(1 - e⁻ᵗ/ᶜ), where a and c are positive constants, t is measured in seconds, and v has units of m/s. (Source: Joe Keller, A Theory of Competitive Running, Physics Today, 26, Sep 1973)

b. Using the velocity in part (a) and assuming s(0) = 0, find the position function s(t), for t ≥ 0.

Textbook Question

Projection sensitivity

According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.

b. Suppose the actual growth rate is instead 0.7%. What are the resulting doubling time and projected 2050 population?