Question

A community swimming pool starts with 250 gallons of water. Julian starts filling the pool at a rate of 500 gallons per hour.
We can connect the points with a line to show the relationship between gallons of water and time. Is the number of gallons of water in the pool a function of the time? In other words, does the number of gallons in the pool depend on the amount of time that has gone by?
t
g
0
1
2
3
4
5
6
7
8
9
10
0
250
500
750
1,000
1,250
1,500
1,750
2,000
2,250
2,500
Time in hours
Gallons of water in the pool
The number of gallons of water in the pool
a function of the time because the number of gallons of water in the pool
on the amount of time that has gone by.

Answer 1

Yes, the number of gallons of water in the pool is a function of time. Specifically, the relationship can be described as follows:

Let \( t \) represent time in hours, and \( g(t) \) represent the number of gallons of water in the pool at time \( t \). The initial amount of water in the pool is 250 gallons, and Julian adds water at a rate of 500 gallons per hour.

The relationship can be expressed with the following function:

\[
g(t) = 250 + 500t
\]

This function indicates that the number of gallons of water in the pool \( g(t) \) depends on the amount of time \( t \), where:

- At \( t = 0 \) (the starting time), \( g(0) = 250 + 500(0) = 250 \) gallons.
- At \( t = 1 \), \( g(1) = 250 + 500(1) = 750 \) gallons.
- At \( t = 2 \), \( g(2) = 250 + 500(2) = 1,250 \) gallons.
- And so on.

Since for each value of \( t \) (time), there is exactly one corresponding value of \( g(t) \) (gallons of water), we can confirm that the number of gallons in the pool is indeed a function of time.

Thus, the number of gallons of water in the pool **depends** on the amount of time that has gone by, making it a function of time.