https://file.aiquickdraw.com/mathbot/file/1726336227259ibbf57n6.jpg

Let's break down each part of the question and solve it based on the graph.

### Part (a): Choosing Between Sine and Cosine
Looking at the graph, the function starts at its maximum value \(I = 1\) when \(t = 0\), which suggests that it is more suited to be modeled by a **cosine function**. This is because a cosine wave reaches its peak at \(t = 0\), whereas a sine wave would start at zero.

- **Explanation**: A cosine function has the form \(I(t) = A \cos(Bt + C)\). For this graph, since the wave starts at a maximum, it aligns well with the behavior of a cosine function, where \(C = 0\) would model a peak at \(t = 0\).

### Part (b): Finding the Equation of the Function
We need to find the equation of the cosine function that fits this graph. A general cosine equation has the form:

\[
I(t) = A \cos(Bt + C) + D
\]

Where:
- \(A\) is the amplitude (the maximum value of the wave),
- \(B\) affects the period (how often the wave repeats),
- \(C\) is the phase shift (horizontal shift),
- \(D\) is the vertical shift.

From the graph:
- The **amplitude** \(A\) is 1, as the graph oscillates between \(1\) and \(-1\).
- The graph completes one cycle from \(t = 0\) to \(t = 4\), meaning the **period** is 4 seconds. The period of a cosine function is given by \( \frac{360^\circ}{B} \), so:
  \[
  \frac{360^\circ}{B} = 4 \quad \Rightarrow \quad B = \frac{360^\circ}{4} = 90^\circ
  \]
- There is no **phase shift** (\(C = 0\)) because the wave starts at its maximum at \(t = 0\).
- There is no **vertical shift** (\(D = 0\)) because the graph oscillates around the horizontal axis.

Thus, the equation of the function is:

\[
I(t) = \cos(90^\circ t)
\]

### Part (c): Tapping to the Maximum Intensity
The student taps their foot every time the sound reaches its most intense part, which corresponds to the maximum of the cosine function (i.e., when \(I = 1\)).

From the equation, we know the function reaches its maximum at the start of every period. The period is 4 seconds, so the student taps their foot every 4 seconds.

Now, let's calculate how many times they will tap in 14 minutes:
- 14 minutes = \(14 \times 60 = 840\) seconds.
- Number of taps = \( \frac{840}{4} = 210\) taps.

### Summary:
1. **(a)**: A cosine function is better suited because the graph starts at its maximum.
2. **(b)**: The equation of the function is \(I(t) = \cos(90^\circ t)\).
3. **(c)**: The student will tap their foot 210 times in 14 minutes.

Would you like further details or explanations?

Here are 5 follow-up questions:
1. What would change if the graph had a vertical shift?
2. How does the phase shift affect the equation of a cosine wave?
3. Could this graph be modeled by a sine function with adjustments?
4. How would you calculate the frequency of this wave?
5. What happens to the period if the frequency doubles?

**Tip**: Always start by identifying key properties such as amplitude, period, and phase shift when modeling periodic functions.

Sounds around us create pressure waves. Our ears interpret the amplitude and frequency of these waves to make sense of the sounds. A speaker is set to create a single tone, and the graph below shows how the pressure intensity (I) of the tone, relative to atmospheric pressure, changes over t seconds: (a) State whether a sine or cosine function would be more suited for modelling this graph. Explain your answer. (b) Find the equation of the function for the intensity I after t seconds. (c) As a brain exercise, a music student must tap their foot every time they hear the most intense part of the sound. How many times will the student tap their foot in 14 minutes?

Learn how to model sound intensity using a cosine function by interpreting the amplitude, period, and graph of sound waves. This problem involves identifying the correct trigonometric function, deriving an equation, and solving a practical problem involving periodic motion. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation