A small outdoor pool is filled with water from a hose. Before water is added from the hose, Sasha notices that the pool has 3 gallons of water in it from the rain. When the hose is turned on, water is added at a constant rate of 2 gallons per minute.
Sasha says that the amount of water in the pool is proportional to the time the hose has been turned on because water flows from the hose at a constant rate. Do you agree with Sasha? 


No, I do not agree with Sasha's statement. While it is true that water flows from the hose at a constant rate of 2 gallons per minute, the total amount of water in the pool is not proportional to the time the hose has been turned on. 

Here's why: proportionality implies a direct relationship of the form \( y = kx \), where \( y \) is the total amount of water, \( x \) is the time, and \( k \) is a constant. This relationship would mean that when time is zero, the total amount of water should also be zero. However, in this case, the pool already contains 3 gallons of water before the hose is turned on. 

Instead of a proportional relationship, the correct relationship is linear but not proportional. The total amount of water \( W(t) \) in the pool as a function of time \( t \) (in minutes) can be expressed by the equation:
\[
W(t) = 2t + 3
\]
Here, the 2 represents the constant rate of water flow (gallons per minute), and the 3 represents the initial 3 gallons of water in the pool. Since this equation has a non-zero y-intercept (3 gallons), the relationship is linear, not proportional.

### Would you like more details on the differences between proportional and linear relationships?

#### 5 Related Questions:
1. What is the difference between proportional and linear relationships?
2. How can you graph the relationship between time and the amount of water in the pool?
3. What would the equation look like if there was no water in the pool initially?
4. How long would it take for the pool to reach a certain amount of water, say 20 gallons?
5. How would the equation change if the water flow rate increased?

#### Tip: Always check for a non-zero starting value when determining proportionality in relationships.

A small outdoor pool is filled with water from a hose. Before water is added from the hose, Sasha notices that the pool has 3 gallons of water in it from the rain. When the hose is turned on, water is added at a constant rate of 2 gallons per minute. Sasha says that the amount of water in the pool is proportional to the time the hose has been turned on because water flows from the hose at a constant rate. Do you agree with Sasha?

Explore the difference between linear and proportional relationships using a pool water problem where water flows at 2 gallons per minute, starting with 3 gallons from rain. Learn the equation W(t) = 2t + 3, suitable for students in Grades 6-8.

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