Problem 84 - A Graphical Approach to Precalculus with Limits

Question

A slide with a uniform slope is to be built on a level piece of land. There are to be 20 equally spaced supports, with the longest support 15 meters long and the shortest 2 meters long. Find the total length of all the supports.

Step-by-Step Solution

Answer

The total length of all supports is 170 meters.

Step 1: Identify and List the Given Information

The problem provides that there will be 20 supports, the longest support is 15 meters, and the shortest is 2 meters. These supports form an arithmetic sequence where the number of supports (terms) is 20 (\(n = 20\)), the first term (\(a_1\)) is 2 meters, and the last term (\(a_n\)) is 15 meters.

Step 2: Recall the Formula for the Sum of an Arithmetic Sequence

The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]where \(a_1\) is the first term, \(a_n\) is the last term, and \(n\) is the total number of terms.

Step 3: Plug the Values into the Formula

Using the sum formula: \[ S_{20} = \frac{20}{2} \times (2 + 15) \]Substitute \(n = 20\), \(a_1 = 2\), and \(a_n = 15\).

Step 4: Calculate the Total Length of the Supports

Calculate the expression from the formula: \[ S_{20} = 10 \times 17 = 170 \]So, the total length of all the supports is 170 meters.

Key Concepts

Sum of Arithmetic Series

An arithmetic series is formed when you add up the terms of an arithmetic sequence. An arithmetic sequence has terms set in a pattern where each term increases or decreases by a fixed amount, known as the common difference. The sum of this series, known as the sum of an arithmetic series, can be calculated using a specific formula. When dealing with an arithmetic series, the primary goal is often to find the total of all included numbers. This is especially useful when dealing with evenly spaced items, like in our exercise where supports were being used. To calculate this sum, you first determine the number of terms you have ( )n). Then, you identify both the first term ( )a_1) and the last term ( )a_n) of your sequence. With these, you can use the sum formula to find the entire sum of the series with ease.

Arithmetic Series Formula

The arithmetic series formula provides a direct way to calculate the total of all numbers in an arithmetic sequence over a specified number of terms. This formula is:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]This formula is quite intuitive when you break it down:

• The term \(n\) represents the total number of terms in your sequence, which is necessary to understand how many elements you are adding up.
• The terms \(a_1\) and \(a_n\) represent the first and last numbers in your series, respectively. Adding the first and last terms is essential because it helps calculate an average for all pairs of terms.
• Finally, multiplying the average by half of the term count (\(n/2\)) gets the total sum for the sequence.

This formula is incredibly useful as it eliminates the tedious task of manually adding every individual number in the series, which is particularly beneficial when dealing with large datasets.

Arithmetic Progression

An arithmetic progression is a series of numbers where each term after the first is formed by adding a constant to the preceding number. This constant is referred to as the common difference. These sequences create a linear pattern that's both predictable and straightforward to work with. To identify an arithmetic progression, you need to look for this constant difference. For example, if you have a series like 3, 7, 11, 15, you can quickly spot that the common difference is 4 since each number increases by 4 from the previous. Key features of an arithmetic progression include:

• The sequence can either increase (as with positive common differences) or decrease (with negative common differences).
• It allows easy computation of the terms and sum, due to its structured pattern.

Because of these attributes, arithmetic progressions are commonplace in a variety of mathematical and real-world applications, such as financial calculations and in engineering projects like the one outlined in the exercise.