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------------------------------------------------ Question 4: Write a thesis for each of the following problems.

a) A string line is tied around a balance beam and hung so that the balance beam is perfectly balanced. One cup is placed at the right end of the beam. Another cup is placed on the left side of the beam. The one on the left is closer to the middle than the right cup. Two cubes are placed in the cup on the right. The beam now tips downward on the right side.

How many cubes should be placed in the left cup in order to balance the beam? Circle your answer. (1 mark)

To solve the problem of balancing a beam with cups at different distances from the center, we need to think about the concept of torque. Torque is the product of force and distance, and for the beam to balance, the torque on both sides must be equal.

Here's how we can find the solution step-by-step:

1. Understand the Setup:
- There are two cups on the beam: one on the left and one on the right.
- The right cup has two cubes.
- The left cup is closer to the center compared to the right cup.

2. Define Key Terms:
- Let’s assume that one cube has a weight of 1 unit for simplicity.
- Let `d_left` represent the distance from the center to the left cup.
- Let `d_right` represent the distance from the center to the right cup.

3. Balancing the Beam:
- The beam is balanced when the torque (force × distance) on both sides is equal.
- Since the right cup has 2 cubes, the torque on the right side is `2 d_right`.

4. Setting up the Equation:
- We need to find how many cubes, let's call this number `n`, are required in the left cup to balance the beam.
- The torque on the left side is then `n d_left`.
- For balance: `n d_left = 2 d_right`.

5. Solve for `n`:
- Rearrange the equation to solve for `n`:
[tex]\[
n = \frac{2 \times d\_right}{d\_left}
\][/tex]

6. Considering Distance Relation:
- The problem states that the left cup is closer to the center than the right cup. Without specific distances, we assume a typical case where `d_left` is half of `d_right`:
[tex]\[
d\_left = 0.5 \times d\_right
\][/tex]

7. Calculate `n`:
- Substitute `d_left = 0.5 * d_right` into the equation:
[tex]\[
n = \frac{2 \times d\_right}{0.5 \times d\_right} = 4
\][/tex]

So, 4 cubes should be placed in the left cup in order to balance the beam.