Problem 59 - College Algebra

Question

A boulder rolled down a mountain, traveling 6 feet in the fi st second. Each successive second, its distance increased by 8 feet. How far did the boulder travel after 10 seconds?

Step-by-Step Solution

Answer

The boulder traveled 420 feet after 10 seconds.

Step 1: Understanding Initial Conditions

The boulder travels 6 feet in the first second. Let's denote this initial distance by \( d_1 = 6 \) feet.

Step 2: Identify the Pattern of Increase

Each subsequent second, the distance increases by 8 feet. Hence, the distance for the \( n \)-th second can be found using the formula \( d_n = d_1 + (n-1) \cdot 8 \). This represents an arithmetic sequence where each term after the first is obtained by adding 8 to the previous term.

Step 3: Calculating Total Distance after 10 Seconds

To find the total distance traveled after 10 seconds, we need the sum of the arithmetic sequence from \( n = 1 \) to \( n = 10 \). The formula for the sum \( S_n \) of an arithmetic sequence is \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( a_n \) is the last term in the sequence. Calculate \( a_{10} = 6 + (10-1) \cdot 8 = 78 \).

Step 4: Apply the Sum Formula

Using the total sum formula, we plug in \( n = 10 \), \( a_1 = 6 \), and \( a_{10} = 78 \): \( S_{10} = \frac{10}{2} (6 + 78) = 5 \cdot 84 = 420 \) feet. This is the total distance traveled by the boulder after 10 seconds.

Key Concepts

Initial Conditions

When working with arithmetic sequences, it's crucial to first understand the initial conditions. In the case of the rolling boulder, the initial conditions refer to the distance it traveled in the very first second. Here, the boulder traveled 6 feet in that initial second. This distance is a foundational piece of information and is often represented with the notation \( d_1 = 6 \) feet.
This initial measurement sets the groundwork for the pattern that unfolds over subsequent seconds. By grasping the initial conditions, you'll have a clear point from which the sequence starts.

Pattern of Increase

The core concept in understanding an arithmetic sequence is identifying the pattern of increase. In this scenario, after the first second, the boulder's travel distance increases by a steady and consistent 8 feet each second. This kind of regularity is key to defining arithmetic sequences.
The formula to determine the distance traveled in the \( n \)-th second is \( d_n = d_1 + (n-1) \times 8 \).

• Suppose \( n = 3 \), the third second's distance will be \( 6 + 2 \times 8 = 22 \) feet.
• This calculation reflects the predictable progression inherent to arithmetic sequences.

Recognizing this linear pattern will help you calculate distances or other elements efficiently.

Sum Formula

Once the pattern and initial conditions are understood, the next step involves calculating the total distance covered over a period. This is calculated using the sum formula for arithmetic sequences. The formula is: \[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \] where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.

For the boulder rolling downhill for 10 seconds:

• First term \( a_1 = 6 \) feet.
• Last term \( a_{10} = 78 \) feet.
• The sum is \( S_{10} = \frac{10}{2} \cdot (6 + 78) = 420 \) feet.

Understanding the sum formula allows for quick calculations of total areas covered by constant growth patterns.

Distance Calculation

The culmination of the process involves using the derived sum to determine the boulder's total distance traveled over the given timeframe. Using the sum formula, the total distance covered from the first to the tenth second is calculated as 420 feet.
This final result represents the comprehensive sum of the boulder's sequential travel, reflecting each momentary increase over 10 seconds.
The formula involves simple arithmetic but illustrates foundational principles in sequences and series. Mastery of distance calculation through arithmetic sequences equips one to solve similar problems efficiently.