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 7.2.32b

Chapter 7, Problem 7.2.32b

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.

b. After how many years is the value of the machine 10% of its original value?

Verified step by step guidance

Identify the initial value of the machine, which is $2.5$ million. Let this be denoted as $V_0 = 2.5$ million.

Since the machine depreciates by 6.8% each year, the value after each year is 93.2% (i.e., $100\% - 6.8\% = 93.2\%$) of the previous year's value. This means the value after $n$ years can be modeled by the exponential decay formula: $V_n = V_0 \times (0.932)^n$.

We want to find the number of years $n$ such that the value $V_n$ is 10% of the original value $V_0$. This gives the equation: $V_0 \times (0.932)^n = 0.10 \times V_0$.

Divide both sides of the equation by $V_0$ to simplify: $(0.932)^n = 0.10$.

To solve for $n$, take the natural logarithm of both sides: $\ln((0.932)^n) = \ln(0.10)$. Using the logarithm power rule, this becomes $n \times \ln(0.932) = \ln(0.10)$. Finally, solve for $n$ by dividing: $n = \frac{\ln(0.10)}{\ln(0.932)}$.

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 Decay

Exponential decay describes a process where a quantity decreases by a fixed percentage over equal time intervals. In this problem, the machine's value decreases by 6.8% each year, meaning its value is multiplied by 0.932 annually. Understanding this helps model the depreciation as a decreasing exponential function.

Exponential Decay Formula

The exponential decay formula is V(t) = V_0 * (1 - r)^t, where V_0 is the initial value, r is the decay rate, and t is time in years. This formula allows calculation of the machine's value after any number of years, which is essential to find when the value reaches 10% of the original.

Solving Exponential Equations Using Logarithms

To find the time t when the value reaches a certain level, we solve equations of the form (1 - r)^t = desired fraction. Taking logarithms on both sides allows isolating t, enabling calculation of the number of years until the machine's value is 10% of its original.

Recommended video:

5:47

Solving Exponential Equations Using Logs

Related Practice

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

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?

Textbook Question

Definitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x² - y² = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as

x = cosh t = (eᵗ + e⁻ᵗ) / 2 and y = sinh t = (eᵗ - e⁻ᵗ) / 2.

b. In Chapter 8, the formula for the integral in part (a) is derived:

∫ √(z² − 1) dz = (z/2)√(z² − 1) − (1/2) ln|z + √(z² − 1)| + C.

Evaluate this integral on the interval [1, x], explain why the absolute value can be dropped, and combine the result with part (a) to show that:

t = ln(x + √(x² − 1)).

Textbook Question

Many formulas There are several ways to express the indefinite integral of sech x.

b. Show that ∫ sech x dx = sin⁻¹ (tanh x) + C. (Hint: Show that sech x = sech² x / √(1 − tanh² x) and then make a change of variables.)

Textbook Question

Power lines A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve ƒ(x) = a cosh x/a. Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = ±50.

c. Use your answer in part (b) to find a, and then compute the length of the power line.