To determine how many overtones exist between 40 Hz and 180 Hz for a stretched string, we first need to understand the relationship between the fundamental frequency and the overtones produced by the string. ### Step 1: Identify the Fundamental Frequency The lowest note emitted by the stretched string is called the fundamental frequency (f₁). In this case, the fundamental frequency is given as: - **f₁ = 40 Hz** ### Step 2: Understand the Harmonics For a stretched string, the frequencies of the harmonics (or overtones) can be calculated using the formula: - **fₙ = n * f₁** where: - **fₙ** is the frequency of the nth harmonic, - **n** is the harmonic number (1 for the fundamental, 2 for the first overtone, 3 for the second overtone, etc.), - **f₁** is the fundamental frequency. ### Step 3: Calculate the Frequencies of the Overtones Using the fundamental frequency of 40 Hz, we can calculate the frequencies of the first few harmonics: - **1st Harmonic (Fundamental)**: - f₁ = 1 * 40 Hz = 40 Hz - **2nd Harmonic (1st Overtone)**: - f₂ = 2 * 40 Hz = 80 Hz - **3rd Harmonic (2nd Overtone)**: - f₃ = 3 * 40 Hz = 120 Hz - **4th Harmonic (3rd Overtone)**: - f₄ = 4 * 40 Hz = 160 Hz - **5th Harmonic (4th Overtone)**: - f₅ = 5 * 40 Hz = 200 Hz ### Step 4: Identify the Overtones Between 40 Hz and 180 Hz Now, we need to find out how many of these harmonics fall between 40 Hz and 180 Hz: - **f₁ = 40 Hz** (not counted as an overtone) - **f₂ = 80 Hz** (1st overtone) - **f₃ = 120 Hz** (2nd overtone) - **f₄ = 160 Hz** (3rd overtone) - **f₅ = 200 Hz** (not counted as it exceeds 180 Hz) ### Conclusion: Count the Overtones The overtones that fall between 40 Hz and 180 Hz are: - 80 Hz (1st overtone) - 120 Hz (2nd overtone) - 160 Hz (3rd overtone) Thus, there are **3 overtones** between 40 Hz and 180 Hz. ### Final Answer The correct option is **B. 3**. ### Explanation of Other Options - **A. 4**: This option is incorrect because there are only three overtones (80 Hz, 120 Hz, and 160 Hz) below 180 Hz. - **C. 2**: This option is incorrect as it undercounts the number of overtones; there are three, not two. - **D. 1**: This option is also incorrect as it significantly undercounts the number of overtones; there are three overtones, not just one. ### Revision Summary - The fundamental frequency of the string is 40 Hz. - The frequencies of the overtones are multiples of the fundamental frequency. - The overtones between 40 Hz and 180 Hz are 80 Hz, 120 Hz, and 160 Hz. - The total number of overtones in this range is 3.