To estimate the amount of quarts of water left after 30 minutes based on the given data, we can use linear interpolation. Here is a detailed, step-by-step solution:
1. Given Data:
- Time [tex]\( T \)[/tex] in hours: [tex]\( 0, 2, 4 \)[/tex]
- Quarts of water [tex]\( Q \)[/tex] at those times: [tex]\( 80, 45, 40 \)[/tex]
2. Convert 30 minutes to hours:
- 30 minutes is equal to [tex]\( \frac{30}{60} = 0.5 \)[/tex] hours.
3. Identify appropriate intervals:
- Since 0.5 hours is between 0 hours and 2 hours, we'll use the points for time [tex]\( T = 0 \)[/tex] hours and [tex]\( T = 2 \)[/tex] hours for our interpolation.
4. Linear Interpolation Formula:
The linear interpolation formula is:
[tex]\[
Q(T) = Q_1 + \frac{(Q_2 - Q_1)}{(T_2 - T_1)} \times (T - T_1)
\][/tex]
where:
- [tex]\( Q_1 \)[/tex] and [tex]\( Q_2 \)[/tex] are the quarts of water at times [tex]\( T_1 \)[/tex] and [tex]\( T_2 \)[/tex] respectively,
- [tex]\( T \)[/tex] is the time at which you're estimating the quarts of water.
5. Apply the Interpolation:
- Use the points [tex]\( (T_1, Q_1) = (0, 80) \)[/tex] and [tex]\( (T_2, Q_2) = (2, 45) \)[/tex].
- Plug these values along with [tex]\( T = 0.5 \)[/tex] into the formula.
[tex]\[
Q(0.5) = 80 + \frac{(45 - 80)}{(2 - 0)} \times (0.5 - 0)
\][/tex]
[tex]\[
Q(0.5) = 80 + \left( \frac{-35}{2} \right) \times 0.5
\][/tex]
[tex]\[
Q(0.5) = 80 + (-17.5) \times 0.5
\][/tex]
[tex]\[
Q(0.5) = 80 - 8.75
\][/tex]
[tex]\[
Q(0.5) = 71.25
\][/tex]
6. Conclusion:
- After 30 minutes (or 0.5 hours), the estimated quarts of water remaining is [tex]\( 71.25 \)[/tex].
