Problem 58 - Basic Technical Mathematics with Calculus

Question

Solve the given problems.The sound produced by a jet engine was measured at a distance of \(100 \mathrm{m}\) in all directions. The loudness \(d\) of the sound (in decibels) was found to be \(d=70.0+30.0 \cos \theta,\) where the \(0^{\circ}\) line was directed in front of the engine. Calculate \(d\) for \(\theta=54.5^{\circ}\).

Step-by-Step Solution

Answer

The loudness \(d\) at \(\theta = 54.5^{\circ}\) is approximately 87.2 decibels.

Step 1: Understand the Problem

We are given a formula for the loudness of a jet engine at a distance of 100 meters: \( d = 70.0 + 30.0 \cos \theta \). The problem asks us to find the loudness \(d\) when \(\theta = 54.5^{\circ}\).

Step 2: Substitute the Value of \(\theta\)

Substitute \(\theta = 54.5^{\circ}\) into the loudness equation. This will give us \(d = 70.0 + 30.0 \cos 54.5^{\circ}\).

Step 3: Calculate \(\cos 54.5^{\circ}\)

Use a calculator to find \(\cos 54.5^{\circ}\). The cosine of 54.5 degrees is approximately 0.5736.

Step 4: Substitute and Solve

Substitute the cosine value into the equation: \(d = 70.0 + 30.0 \times 0.5736\). Calculate the multiplication and addition to find \(d\).

Step 5: Perform the Calculation

First compute the multiplication: \(30.0 \times 0.5736 = 17.208\). Next, add this to 70.0: \(70.0 + 17.208 = 87.208\). So, \(d = 87.208\) decibels.

Key Concepts

Cosine Function

Trigonometric functions, like the cosine function, are fundamental in mathematics, especially in dealing with phenomena involving waves or oscillations. The cosine function is part of the group of functions known as circular functions. It's one of the essential functions alongside sine and tangent.
Cosine takes an angle as input and returns the horizontal coordinate of a point on the unit circle. In simpler terms, when you imagine a circle, the cosine of an angle is the distance from the center to the side along the x-axis when you move that angle from the center. Here's how you can look at it:

• If the angle is 0 degrees, the cosine value is 1.
• At 90 degrees, the cosine value is 0.
• It starts to decrease from 1 at 0 degrees to 0 at 90 degrees and continues this oscillating pattern.

Cosine values vary between -1 and 1, making it a periodic function repeating values every 360 degrees. In practical applications, these cosine values can be found using calculators, making it easier to use them in equations like the sound measurement equation used in the exercise.

Sound Measurement

Sound measurement involves quantifying sound waves using various parameters. The most common parameters are loudness (measured in decibels), frequency, and duration. Sound travels in waves, and these waves create pressure variations in the air, which we perceive as sound.
The decibel (dB) is a logarithmic unit used to measure sound intensity. Unlike linear measurements, the logarithmic scale represents intensity in exponential steps. This means every increase in 10 dB represents a tenfold increase in sound intensity.
When measuring sound with angles, as in the exercise, different angles can have varied effects on perceived loudness due to directionality. The equation given, \(d = 70.0 + 30.0 \cos \theta\), indicates how loudness changes with the direction of measurement:

• \(\theta = 0^{\circ}\) implies maximum effect in front of the engine, likely the loudest.
• Changes in \(\theta\) indicate directional variance, showing how sound intensity is quieter or louder depending on angle.

Thus, understanding sound measurement is crucial for assessing how sound affects our environment and for engineering solutions to reduce unwanted noise.

Loudness Calculation

Loudness calculation involves using mathematical equations to determine how loud a sound is perceived. In the context of the exercise, we're looking at a specific formula relating to how directional angles change the perceived loudness of a jet engine.
Given the equation: \[ d = 70.0 + 30.0 \cos \theta \]This tells us that the loudness, measured in decibels, changes with direction due to the cosine component. The formula includes:

• A base loudness of 70.0 dB, representing the minimum loudness at all directions.
• An additional component, \(30.0 \cos \theta\), modifying this base loudness based on direction.

To calculate the loudness at a certain angle \(\theta\), as in the example of \(54.5^{\circ}\):

• Find \(\cos 54.5^{\circ}\) using a calculator, approximately 0.5736.
• Substitute this into the equation: \(d = 70.0 + 30.0 \times 0.5736\).
• Perform the arithmetic: \(d = 70.0 + 17.208 = 87.208 \; dB\).

Loudness calculation is essential in contexts where noise control and regulation is important, helping to ensure environments remain within safe and comfortable sound levels.