Question

Several students performed an experiment by dropping marbles that are uniform in size into a jar partially filled with water. They measured the height of the water in the jar after every five marbles. The results are shown in the table.


Based on the table, write an equation that models this situation.

Answer 1

I don't see the table in your message. Could you either paste the measured heights here or tell me two rows (for example, height with 0 marbles and height with 5 marbles)? With that I can give the exact equation.

Meanwhile, here is how you get the model from the table:

1. Decide the independent variable:
- Let M = number of marbles, or
- Let k = number of groups of five marbles (k = M/5).

2. Use two data points from the table, written as (M, H) where H is the water height.
- If your table gives heights after every 5 marbles, you will have points like (0, H0), (5, H5), (10, H10), …

3. Compute the slope (increase in height per marble):
slope per marble = m = (H2 − H1) / (M2 − M1).
If you use two consecutive five-marble entries, slope per five marbles = (H_next − H_prev); then divide by 5 to get per marble.

4. Form the linear equation:
H = m·M + b,
where b is the initial height (H when M = 0) or find b by plugging one point: b = H1 − m·M1.

5. (Alternative using groups of five)
If you prefer k = number of five-marble groups, slope per group = (H_next − H_prev), and
H = m_group·k + b, with m_group = increase in H per 5 marbles.

Example (illustration):
- Suppose H(0) = 2.0 cm and H(5) = 3.0 cm.
slope per marble m = (3.0 − 2.0) / (5 − 0) = 1.0/5 = 0.2 cm/marble.
Equation: H = 0.2·M + 2.0.

Send the table or two (or more) points and I will compute the exact equation for you.