Certainly! Let's go through the process step-by-step to find the x-intercept and understand its meaning in this scenario.
1. Understanding the Data:
- We have a table that shows the amount of water leaking from an inflatable pool over time. We can interpret the given table to understand that:
- At [tex]\( t = 0 \)[/tex], the amount of water is 30 liters.
- At [tex]\( t = 1 \)[/tex], the amount of water is 25 liters.
- At [tex]\( t = 2 \)[/tex], the amount of water is 20 liters.
- At [tex]\( t = 3 \)[/tex], the amount of water is 15 liters.
2. Determining the Rate of Change (Slope):
- The data represents a linear relationship between time and the amount of water. We need to calculate the rate at which the water is leaking.
- The slope of the line ([tex]\( m \)[/tex]) can be calculated using any two points from the table, such as (0, 30) and (1, 25).
[tex]\[
m = \frac{ \Delta y }{ \Delta x } = \frac{ 25 - 30 }{ 1 - 0 } = \frac{ -5 }{ 1 } = -5 \text{ liters per minute}
\][/tex]
3. Finding the x-intercept:
- The x-intercept of a line is the point where the line crosses the x-axis, which corresponds to when the amount of water in the pool reaches 0 liters.
- Starting from the equation of a line in slope-intercept form [tex]\( y = mx + b \)[/tex]:
- Here [tex]\( m \)[/tex] is the slope ([tex]\( -5 \)[/tex]).
- [tex]\( b \)[/tex] is the y-intercept, which is 30 (the initial amount of water at [tex]\( t = 0 \)[/tex]).
- We need to find [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex]:
[tex]\[
0 = -5x + 30
\][/tex]
[tex]\[
5x = 30
\][/tex]
[tex]\[
x = \frac{ 30 }{ 5 } = 6 \text{ minutes}
\][/tex]
4. Interpretation:
- The x-intercept [tex]\( x = 6 \)[/tex] represents the time it takes for the pool to completely empty out, assuming the rate of leakage remains constant.
Thus, in this scenario, the x-intercept ([tex]\( 6, 0 \)[/tex]) indicates that it will take 6 minutes for the pool to be entirely empty of water.
