Problem 30 - Technical Mathematics with Calculus

Question

Atmospheric Pressure: The pressures measured at 1 -mi intervals above sea level form a GP, with each value smaller than the preceding by a factor of \(0.819 .\) If the pressure at sea level is 29.92 in. Hg, find the pressure at an altitude of 5 mi.

Step-by-Step Solution

Answer

The pressure at an altitude of 5 mi is approximately \( 29.92 \times (0.819)^4 \) in. Hg.

Step 1: Identify the Type of Sequence

Recognize that the pressures form a geometric progression (GP). In a GP, each term after the first is found by multiplying the previous term by a constant called the common ratio.

Step 2: Determine the Common Ratio

The problem states that each value is smaller than the preceding by a factor of 0.819, which means the common ratio (r) of the GP is 0.819.

Step 3: Write the General Formula for the nth Term of a GP

The nth term of a GP can be found using the formula: \( a_n = a_1 \times r^{(n-1)} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

Step 4: Apply the Formula to Find the Pressure at 5 mi

Plug in the values: sea level pressure (\( a_1 = 29.92 \) in. Hg), the common ratio (\( r = 0.819 \)), and the altitude in miles (n = 5). So, the pressure at an altitude of 5 mi is \( a_5 = 29.92 \times 0.819^{(5-1)} \).

Step 5: Calculate the Fifth Term of the GP

Calculate \( a_5 \) by raising the common ratio to the fourth power and then multiplying it by the first term: \( a_5 = 29.92 \times 0.819^4 \).

Step 6: Simplify the Calculation

Use a calculator to find \( 0.819^4 \) and multiply by 29.92 to find the pressure at 5 mi.

Key Concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted on a surface by the air above it in the atmosphere. At sea level, this pressure is at its highest because the density of the air is greater. As we ascend to higher altitudes, the air becomes less dense, resulting in a decrease in atmospheric pressure. This is an essential concept in various scientific fields, including meteorology, aviation, and environmental science. It's measured in units such as inches of mercury (in. Hg) or millibars. The decrease in atmospheric pressure at higher altitudes can be mathematically represented by a geometric progression, where each pressure value is successively reduced by a recurring factor.

Common Ratio

In a geometric progression (GP), the common ratio is the constant factor that each term of the sequence is multiplied by to get the next term. For example, if you have a sequence of numbers where each one is 0.819 times the previous one, then your common ratio is 0.819. This concept is key to understanding GP as it determines the pattern of change from one term to the next. Whether the sequence is increasing or decreasing, and at what rate, is dependent entirely on the common ratio. A common ratio greater than 1 means the sequence is increasing, while a ratio between 0 and 1, like in our atmospheric pressure example, indicates a decreasing sequence.

Nth Term Formula

The nth term formula is vital when working with any regular sequence, but it is particularly crucial in geometric progressions. It allows you to quickly find any term in the sequence without needing to calculate all the previous terms. In the context of our atmospheric pressure example, the nth term formula is expressed as \( a_n = a_1 \times r^{(n-1)} \), where \( a_n \) is the nth term, \( a_1 \) is the first term, and \( r \) is the common ratio. Using this formula, you can determine the atmospheric pressure at any given altitude (or term in our sequence) by knowing the initial pressure at sea level and the common ratio.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers following a specific rule that allows you to predict subsequent values. Sequences come in various types such as arithmetic, geometric, or more complex ones like Fibonacci. The geometric progression is a type of sequence where each term after the first is produced by multiplying the previous one by a fixed, non-zero number known as the common ratio. This concept is immensely important in mathematics because it helps us model and solve real-world problems, such as calculating compounding interest, population growth, or, as in our example, changes in atmospheric pressure with altitude.