Problem 37 - Calculus Volume 1

Question

To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1 and adds \(\frac{1}{10} \mathrm{mi}\) to his daily routine each week, what is the total mileage on Joe's shoes after 25 weeks?

Step-by-Step Solution

Answer

Joe runs a total of 205 miles after 25 weeks.

Step 1: Understanding the Problem

Joe starts running 1 mile each day in the first week and increases his mileage by \(\frac{1}{10}\) mile each week over a period of 25 weeks. We need to calculate the total distance Joe has run after these 25 weeks.

Step 2: Calculate Weekly Mileage Increase

Joe increases his daily mileage by \(\frac{1}{10}\) mile each week. By week 2, he runs \(1 + \frac{1}{10} = 1.1\) miles each day. By week 3, it's \(1.2\) miles each day, continuing this pattern.

Step 3: Calculate Total Mileage for Each Week

To find the total mileage run each week, multiply the weekly daily mileage by 7 (days in a week). For example, in week 1: \(1 \text{ mile/day} \times 7 \text{ days/week} = 7 \text{ miles}\). Continue this process for each subsequent week, adding \(\frac{1}{10} \text{ mile/day/week}\).

Step 4: Use Arithmetic Series Formula

The increase in weekly mileage forms an arithmetic series. The formula for the sum of an arithmetic series is \(S_n = n/2 \times (a + l)\), where \(n\) is the total number of terms, \(a\) is the first term, and \(l\) is the last term. Here, \(a = 7\) miles (week 1 total mileage), \(l = 9.4\) miles (week 25 total mileage), and \(n = 25\) weeks.

Step 5: Calculate the Total Mileage After 25 Weeks

Using the arithmetic series formula: \(S_{25} = \frac{25}{2} \times (7 + 9.4) = \frac{25}{2} \times 16.4 = 205\). Therefore, the total mileage on Joe's shoes after 25 weeks is 205 miles.

Key Concepts

Understanding Arithmetic Sequences

A sequence is an ordered list of numbers, where each number is called a term. In Joe's exercise routine, the daily mileage he runs each week increases by a fixed amount, forming an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In this scenario, Joe increases his daily mileage by \( \frac{1}{10} \) mile each week.

This consistent increase means that each week's daily distance is part of this arithmetic sequence. Identifying that Joe's routine is an arithmetic sequence allows us to use mathematical formulas designed to handle such patterns, greatly simplifying the calculation of the total mileage over several weeks.

Solving the Mileage Problem

When tackling problems like Joe's running routine, breaking it down step-by-step can be very helpful. Initially, you recognize the exercise scenario and determine what is being asked. Here, you want to calculate the total mileage Joe accumulates after 25 weeks, given he starts with running 1 mile daily and increases by \( \frac{1}{10} \) mile each week.

Here's a step-by-step approach to solving this:

• Determine the arithmetic sequence for daily mileage: Joe begins with 1 mile and increases it by \( 0.1 \) miles each week. So by week 2, he runs 1.1 miles, and 1.2 miles by week 3, etc.
• Multiply daily mileage by the number of days in a week (7) to find the weekly mileage: for example, 7 miles in week 1, 7.7 miles in week 2, and so on.
• Add the weekly distances over 25 weeks. Here, using the arithmetic series formula \( S_n = \frac{n}{2} \times (a + l) \) simplifies this significantly.

This systematic breakdown aids in piecing together the problem and finding a solution efficiently.

Calculating Total Distance

Finally, we get to calculating the total distance covered, which ties in nicely with understanding an arithmetic sequence. In exercises like these, a common challenge is to calculate the sum of a series of numbers efficiently. The formula for the sum of an arithmetic series \( S_n = \frac{n}{2} \times (a + l) \) is vital here.

Using this formula:

• The total number of weeks, \( n \), is 25.
• The first term of the mileage series, \( a \), is 7 miles (week 1).
• The last term, \( l \), is 9.4 miles (week 25).

Plug these values into the formula to get \( S_{25} = \frac{25}{2} \times (7 + 9.4) = 205 \). Hence, Joe's total mileage after 25 weeks is 205 miles. This insight showcases how understanding and applying arithmetic series concepts are crucial in solving this distance calculation problem.