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Let's break down the problem step by step based on the image.

### Given Formula:
The frequency \( f \) of the sound emitted by the string of a guitar in Hertz (Hz) is given by:
\[
f = 10 \times \sqrt{t}
\]
where:
- \( t \) is the tension of the string in Newtons (N).

### Task 1: Calculate the frequency for given tensions
You are asked to find the frequency for three different tensions:
1. \( t = 100 \, N \)
2. \( t = 225 \, N \)
3. \( t = 400 \, N \)

#### For \( t = 100 \, N \):
\[
f = 10 \times \sqrt{100} = 10 \times 10 = 100 \, \text{Hz}
\]

#### For \( t = 225 \, N \):
\[
f = 10 \times \sqrt{225} = 10 \times 15 = 150 \, \text{Hz}
\]

#### For \( t = 400 \, N \):
\[
f = 10 \times \sqrt{400} = 10 \times 20 = 200 \, \text{Hz}
\]

---

### Task 2: Find the tension for a frequency of 440 Hz (La₃)
To obtain the note **La₃** with a frequency of \( f = 440 \, \text{Hz} \), we need to find the tension \( t \).

We use the formula \( f = 10 \times \sqrt{t} \):
\[
440 = 10 \times \sqrt{t}
\]
Dividing both sides by 10:
\[
\sqrt{t} = 44
\]
Now, squaring both sides:
\[
t = 44^2 = 1936 \, N
\]

---

### Task 2a: Explanation of Romane's reasoning
Romane claims: "In this case, the square root of \( t \) is equal to 44." This means that Romane has correctly applied the formula \( f = 10 \times \sqrt{t} \) and set \( f = 440 \, \text{Hz} \). By dividing 440 by 10, Romane found that \( \sqrt{t} = 44 \), leading to the correct value of the tension.

---

Would you like to go over any particular step in more detail or have additional questions? Here are some related questions for further practice:

1. What would the frequency be for a tension of \( 150 \, N \)?
2. How does the frequency change if the tension is quadrupled?
3. What is the tension required to produce a frequency of \( 300 \, \text{Hz} \)?
4. How is the frequency related to the square root of the tension mathematically?
5. Why does increasing tension lead to higher frequencies in musical instruments?

**Tip**: In musical instruments, the pitch (frequency) is directly related to the tension in the strings: increasing tension leads to higher pitch.

La fréquence f du son émis par la corde d'une guitare, en hertz (Hz), est f = 10 × √t, où t est la tension de la corde, en newton (N). Calculer la fréquence pour des tensions de 100 N, 225 N, et 400 N. Déterminer également la tension pour obtenir une fréquence de 440 Hz.

Learn how to calculate the frequency emitted by a guitar string for different tensions using the formula f = 10 × √t. Explore examples for tensions of 100 N, 225 N, and 400 N, and determine the tension required for a frequency of 440 Hz.

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