Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.R.95b

Chapter 10, Problem 10.R.95b

Building a tunnel — first scenario
A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.

b.What is the longest tunnel the crew can build at this rate?

Verified step by step guidance

Recognize that the distance dug each week forms a geometric sequence where the first term \(a_1 = 100\) meters and the common ratio \(r = 0.95\) because each week the crew digs 0.95 times the previous week's distance.

Recall that the total length of the tunnel after infinitely many weeks is the sum of an infinite geometric series, which converges when \(|r| < 1\).

Use the formula for the sum of an infinite geometric series: \(S_\infty = \frac{a_1}{1 - r}\), where \(a_1\) is the first term and \(r\) is the common ratio.

Substitute the known values into the formula: \(S_\infty = \frac{100}{1 - 0.95}\) to express the total maximum length of the tunnel.

Interpret this sum as the longest tunnel the crew can build at this decreasing rate of construction.

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Key Concepts

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

Geometric Series

A geometric series is the sum of terms where each term is a constant multiple (common ratio) of the previous one. In this problem, the weekly distances dug form a geometric sequence with ratio 0.95. Understanding how to sum an infinite geometric series helps find the total tunnel length when the digging rate decreases continuously.

Sum to Infinity of a Geometric Series

When the common ratio of a geometric series is between -1 and 1, the series converges to a finite sum. The formula S = a / (1 - r) calculates this sum, where 'a' is the first term and 'r' is the ratio. This concept is essential to determine the maximum tunnel length achievable as the digging rate diminishes over time.

Modeling Real-World Problems with Sequences

Translating a real-world scenario into a mathematical sequence allows for precise analysis. Here, the weekly digging distances decrease by a fixed percentage, forming a geometric sequence. Recognizing this pattern enables the use of sequence formulas to predict total progress and solve practical construction problems.

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