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

Briggs - Calculus: Early Transcendentals 3rd Edition

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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: $n = \frac{\ln(0.10)}{\ln(0.932)}$.

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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.

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Solving Exponential Equations Using Logs

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Textbook Question

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.

b. Find the function that gives the amount of oil consumed between t=0 and any future time t.

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Textbook Question

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.

a. What is the value of the machine after 10 years?

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Textbook Question

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a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.

104

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b. Compute the area of the region described.

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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.

a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remains constant.

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b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?

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