Let A(t) represent the volume of water in a tank t minutes after the tank starts to fill. The rate of change of the volume after t minutes, in liters per minute, is given by r (t) = 0.5·t3- 11·t2+76· t1. Approximate the number of liters of water in the tank 1 minute and 20 seconds after it started filling, using accumulation over time intervals of 40 seconds. 2. Approximate the number of liters of water in the tank 7 minutes and 30 seconds after it started filling, using accumulation over time intervals of 80 seconds.
Answer
🚰 Approximating Water Volume in a Tank Using Rate Function
🔍 Given Rate Function
The rate of change of the volume of water in the tank is:
r(t) = 0.5·t³ – 11·t² + 76·t (liters/minute)📘 Method: Accumulation Over Time Intervals
To find the approximate volume, we divide time into intervals and calculate:
A(t) ≈ Σ [ r(ti) × Δt ]where r(ti) is the rate at the start of the interval and Δt is the width of the interval (in minutes).
📌 Part 1: Approximate Volume at 1 Minute 20 Seconds (1.33 minutes)
Interval Width: 40 seconds = 40/60 = 0.667 minutes
Break [0, 1.33] into 2 intervals: [0, 0.667], [0.667, 1.33]
Step-by-step Calculation:
| Interval | Start Time (min) | r(t) (L/min) | Δt (min) | r(t) × Δt |
|---|---|---|---|---|
| 1 | 0.000 | 0.0 | 0.667 | 0.0 |
| 2 | 0.667 | 0.5*(0.667)^3 – 11*(0.667)^2 + 76*(0.667) ≈ 42.7 | 0.667 | 42.7 × 0.667 ≈ 28.48 |
Total Approximate Volume: A(1.33) ≈ 0 + 28.48 = 28.48 liters
📌 Part 2: Approximate Volume at 7 Minutes 30 Seconds (7.5 minutes)
Interval Width: 80 seconds = 80/60 = 1.333 minutes
Break [0, 7.5] into 6 intervals: [0, 1.333], [1.333, 2.666], …, [6.667, 8.000]
Step-by-step Calculation:
| Interval | Start Time (min) | r(t) (L/min) | Δt (min) | r(t) × Δt |
|---|---|---|---|---|
| 1 | 0.000 | 0.0 | 1.333 | 0.0 |
| 2 | 1.333 | ≈70.0 | 1.333 | ≈93.3 |
| 3 | 2.666 | ≈98.4 | 1.333 | ≈131.1 |
| 4 | 4.000 | ≈86.0 | 1.333 | ≈114.7 |
| 5 | 5.333 | ≈54.6 | 1.333 | ≈72.8 |
| 6 | 6.667 | ≈24.8 | 0.833 | ≈20.7 |
Total Approximate Volume: A(7.5) ≈ 0 + 93.3 + 131.1 + 114.7 + 72.8 + 20.7 = 432.6 liters
✅ Final Results
- At 1 minute 20 seconds: ~28.48 liters
- At 7 minutes 30 seconds: ~432.6 liters
These are left-endpoint Riemann sum approximations. Accuracy improves with smaller interval widths or using midpoint or trapezoidal methods.