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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 7.2.32a
Chapter 7, Problem 7.2.32a

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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1
Identify the initial value of the machine, which is \$2.5 million. This will be our starting amount, denoted as \(V_0 = 2.5\) million.
Understand that the machine depreciates by 6.8% each year based on its current value. This means each year the machine retains 100% - 6.8% = 93.2% of its value from the previous year.
Express the value of the machine after \(n\) years using the exponential decay formula: \(V_n = V_0 \times (1 - r)^n\), where \(r\) is the depreciation rate as a decimal (6.8% = 0.068).
Substitute the known values into the formula: \(V_{10} = 2.5 \times (1 - 0.068)^{10}\) to represent the value after 10 years.
To find the value after 10 years, calculate the expression \(2.5 \times 0.932^{10}\). This will give the depreciated value of the machine after 10 years.

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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 after each year is multiplied by (1 - 0.068). This concept helps model the depreciation over multiple years.
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Depreciation

Depreciation is the reduction in the value of an asset over time, often due to wear and tear or obsolescence. Here, the machine’s value decreases annually by a fixed percentage, which is a common method for calculating tax deductions and understanding asset worth.

Compound Interest Formula

The compound interest formula, V = P(1 - r)^t, calculates the future value of an amount after repeated percentage decreases or increases. In depreciation, it models how the value diminishes year after year, where P is the initial value, r is the depreciation rate, and t is the number of years.
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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.


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

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