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Let's break down and solve each problem step by step.

### Problem 6: Seesaw and Archimedes' Law of the Lever

#### Part (a)
The problem states that the distance \(d\) from the pivot is inversely proportional to the weight \(w\). This means the product of distance and weight is constant:

\[
d \cdot w = k
\]

where \(k\) is a constant of proportionality. So, the function to represent the Law of the Lever is:

\[
d = \frac{k}{w}
\]

or alternatively,

\[
d \cdot w = k
\]

#### Part (b)
We are given the following information:

- Mark weighs 20 kg and is 1.5 m from the pivot.
- Emilie sits 1.25 m from the pivot, and the seesaw is balanced.

Since the seesaw is balanced, the moments on both sides of the pivot must be equal. Therefore, using the Law of the Lever:

\[
\text{Mark's moment} = \text{Emilie's moment}
\]

This means:

\[
20 \times 1.5 = k \times 1.25
\]

Solving for \(k\):

\[
30 = k \times 1.25
\]

\[
k = \frac{30}{1.25} = 24
\]

So, \(k = 24\).

#### Part (c)
To find Emilie's weight \(w_E\), we use the equation for the lever:

\[
d_E \cdot w_E = k
\]

Substitute \(d_E = 1.25\) m and \(k = 24\):

\[
1.25 \cdot w_E = 24
\]

Solving for \(w_E\):

\[
w_E = \frac{24}{1.25} = 19.2 \, \text{kg}
\]

So, Emilie's weight is 19.2 kg.

---

### Problem 7: Inverse Proportionality Verification

We are given a table that shows the number of people and the corresponding cost per person:

| Number of people | 1   | 4    | 8     | 10   | 20  |
|------------------|-----|------|-------|------|-----|
| Cost per person  | 300 | 75   | 37.5  | 30   | 15  |

We need to verify that the number of people \(n\) and the cost per person \(C\) are inversely proportional. Inverse proportionality means:

\[
n \cdot C = \text{constant}
\]

Let's calculate \(n \cdot C\) for each pair:

- For \(n = 1\) and \(C = 300\): \(1 \times 300 = 300\)
- For \(n = 4\) and \(C = 75\): \(4 \times 75 = 300\)
- For \(n = 8\) and \(C = 37.5\): \(8 \times 37.5 = 300\)
- For \(n = 10\) and \(C = 30\): \(10 \times 30 = 300\)
- For \(n = 20\) and \(C = 15\): \(20 \times 15 = 300\)

Since \(n \cdot C\) is constant for all values, the relationship is indeed inversely proportional.

---

Would you like more details on any of these solutions, or do you have any questions?

### Relative Questions:
1. How can the Law of the Lever be applied to complex systems involving multiple objects?
2. Can the constant \(k\) change under different conditions in the seesaw example?
3. How would the solution change if Mark moved further away from the pivot?
4. What other types of physical systems demonstrate inverse proportionality?
5. How would you set up an equation if the number of people and cost were not perfectly inversely proportional?

### Tip:
Inverse proportionality means that as one quantity increases, the other decreases at the same rate, and their product remains constant.

To balance a seesaw, Archimedes' Law of the Lever says that the distance of each person from the pivot must be inversely proportional to their weight. a) Write down a function to represent the Law of the Lever. b) Mark weighs 20 kg and is 1.5 m from the pivot. The seesaw is balanced when Emilie sits 1.25 m from the pivot. Use this information to find k. c) Hence, find Emilie’s weight.

Solve the problem of balancing a seesaw using Archimedes' Law of the Lever, where the distance of a person from the pivot is inversely proportional to their weight. Learn how to calculate a constant k and find the unknown weight with step-by-step solutions. Suitable for Grades 8-10 students.

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