Question

the table shows the height, in centimeters, of the water in a swimming pool at different times since the pool started to be filled.

1. does the height of the water increase by the same amount each minute? explain how you know.
2. does the height of the water increase by the same factor each minute? explain how you know.

the table:
minutes | height
0 | 150
1 | 150.5
2 | 151
3 | 151.5

Explanation:

Sub - Question 1

Step1: Calculate the differences between consecutive heights.

First, find the difference between the height at 1 minute and 0 minute: $150.5 - 150 = 0.5$ cm.
Then, find the difference between the height at 2 minutes and 1 minute: $151 - 150.5 = 0.5$ cm.
Next, find the difference between the height at 3 minutes and 2 minutes: $151.5 - 151 = 0.5$ cm.

Step2: Analyze the differences.

Since the difference in height between each consecutive minute (from 0 - 1, 1 - 2, 2 - 3 minutes) is the same (0.5 cm), the height of the water increases by the same amount each minute.

Step1: Calculate the ratios of consecutive heights.

First, find the ratio of the height at 1 minute to the height at 0 minute: $\frac{150.5}{150}\approx1.0033$.
Then, find the ratio of the height at 2 minutes to the height at 1 minute: $\frac{151}{150.5}\approx1.0033$.
Next, find the ratio of the height at 3 minutes to the height at 2 minutes: $\frac{151.5}{151}\approx1.0033$.

Step2: Analyze the ratios.

Since the ratio of the height of the water at each consecutive minute (from 0 - 1, 1 - 2, 2 - 3 minutes) is approximately the same (about 1.0033), the height of the water increases by the same factor each minute.

Answer:

Yes, the height of the water increases by the same amount (0.5 cm) each minute. We know this by calculating the difference in height between consecutive minutes (150.5 - 150 = 0.5, 151 - 150.5 = 0.5, 151.5 - 151 = 0.5) and seeing that the differences are equal.

Sub - Question 2