Problem 27 - Basic Engineering Mathematics

Question

A drilling machine is to have 6 speeds ranging from \(50 \mathrm{rev} / \mathrm{min}\) to \(750 \mathrm{rev} / \mathrm{min}\). If the speeds form a geometric progression determine their values, each correct to the nearest whole number.

Step-by-Step Solution

Answer

The speeds are 50, 97, 188, 363, 702, and 750 rev/min.

Step 1: Define the Problem

We need to calculate 6 speeds of a drilling machine that form a geometric progression and range from 50 revolutions per minute (rev/min) to 750 rev/min.

Step 2: Recall the Formula for Geometric Progression

In a geometric progression, each term after the first is found by multiplying the previous term by a constant known as the common ratio, denoted as \(r\). For a sequence \(a, ar, ar^2, \ldots\), the \(n\)-th term is given by \(ar^{n-1}\).

Step 3: Setting Up the Equations

Given that the first speed is \(a = 50\) and the last speed is \(ar^5 = 750\), we can set up the equation: \(50r^5 = 750\).

Step 4: Solve for the Common Ratio

Rearrange the equation to solve for \(r^5\): \(r^5 = \frac{750}{50} = 15\). Now take the fifth root to find \(r\): \(r = 15^{1/5}\).

Step 5: Calculate the Common Ratio

By calculating the fifth root, we get \(r \approx 1.933\). This is the common ratio between consecutive speeds.

Step 6: Calculate All Speeds

Using the common ratio, calculate each speed:1. First Speed: \(a = 50\)2. Second Speed: \(ar = 50 \times 1.933 \approx 97\)3. Third Speed: \(ar^2 = 97 \times 1.933 \approx 188\)4. Fourth Speed: \(ar^3 = 188 \times 1.933 \approx 363\)5. Fifth Speed: \(ar^4 = 363 \times 1.933 \approx 702\)6. Sixth Speed: \(ar^5 = 750 (Given)\)Round each to the nearest whole number as needed.

Step 7: Verify the Result

Ensure all calculated speeds form a geometric progression: Each speed calculated should logically follow from multiplying the previous speed by \(r\) and should match with 750 as the last term.

Key Concepts

Common Ratio

In a geometric progression, the key component is the constant multiplier between consecutive terms, known as the common ratio. This ratio, denoted by \( r \), remains consistent throughout the sequence. When you calculate the next term in the sequence, you multiply the current term by the common ratio. For example, if the first term (\( a \)) is 50 and the common ratio is 1.933, the second term can be calculated by multiplying 50 by 1.933.

• The common ratio is essential to maintain the pattern of the sequence.
• It ensures that each term can be derived directly from the preceding one through multiplication.

To solve for the common ratio when the sequence is known, you can use the formula \( r = \left( \frac{a_n}{a} \right)^{1/(n-1)} \), where \( a \) is the first term, \( a_n \) is the last term given, and \( n \) is the total number of terms.

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by the common ratio. This contrast with arithmetic sequences, where terms increase by a constant addition.

• Example of a geometric sequence: 50, 97, 188, 363, 702, 750 represents speeds in revolutions per minute.
• The sequence is defined clearly by its first term and the common ratio.
• A defining characteristic is the exponential growth of terms as you progress through the sequence.

The formula for the \( n \)-th term is \( ar^{n-1} \). Here, \( a \) is the first term, and \( r \) is the common ratio. This formula allows you to determine any term in the sequence without having to calculate all preceding ones.

Mathematical Calculation

Performing mathematical calculations is crucial to solving problems involving geometric sequences. This involves not only applying formulas but also understanding how to manipulate them to find unknown values such as the common ratio or any term in the sequence.

• Calculating the common ratio involves division and extraction of roots: \( r = 15^{1/5} \).
• Each speed in the sequence can be calculated using the geometric progression formula for terms.
• Verification ensures that the calculations are accurate and consistent with the properties of geometric sequences.

Logical steps and operations, such as correctly rounding the results to the nearest whole number, can enhance accuracy in practical situations like determining machine speeds.

Revolutions Per Minute

Revolutions per minute (RPM) is a measure of rotational speed. It's widely used in contexts like machinery, engines, and tools to describe how fast they operate.

• An RPM of 50 means the machine part rotates 50 times in one minute.
• Observing the change in RPM values in geometric progression gives insight into the efficiency and capability of a machine.
• By ensuring the RPMs form a geometric progression, you can achieve a balanced performance across various settings.

In engineering and applied sciences, understanding RPM is crucial for optimizing machinery performance and ensuring that operations remain smooth and effective over different operational modes.