Problem 20 - Applied Calculus

Question

An old rowboat has sprung a leak. Water is flowing into the boat at a rate, \(r(t),\) given in the table. (a) Compute upper and lower estimates for the volume of water that has flowed into the boat during the 15 minutes. (b) Draw a graph to illustrate the lower estimate. $$\begin{array}{l|r|r|r|r} \hline t \text { minutes } & 0 & 5 & 10 & 15 \\ \hline r(t) \text { liters/min } & 12 & 20 & 24 & 16 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Lower estimate: 280 liters. Upper estimate: 300 liters.

Step 1: Understand the Problem

We're given that the rate of water flowing into the boat at specific time intervals is in liters per minute. Our task is to estimate the total volume of water that flows in during the 15 minutes using upper and lower estimates.

Step 2: Determine the Lower Estimate

The lower estimate is calculated using the left Riemann sum method. We use the rate at each beginning interval as an approximation. Therefore, we'll sum the following: - From 0 to 5 minutes: Use rate at 0 min, which is 12 liters/min- From 5 to 10 minutes: Use rate at 5 min, which is 20 liters/min- From 10 to 15 minutes: Use rate at 10 min, which is 24 liters/min.The lower estimate is: \[(5 \times 12) + (5 \times 20) + (5 \times 24) = 60 + 100 + 120 = 280 \text{ liters}\]

Step 3: Determine the Upper Estimate

The upper estimate is calculated using the right Riemann sum method. We use the rate at each end interval as an approximation. Therefore, we'll sum the following: - From 0 to 5 minutes: Use rate at 5 min, which is 20 liters/min- From 5 to 10 minutes: Use rate at 10 min, which is 24 liters/min- From 10 to 15 minutes: Use rate at 15 min, which is 16 liters/min.The upper estimate is: \[(5 \times 20) + (5 \times 24) + (5 \times 16) = 100 + 120 + 80 = 300 \text{ liters}\]

Step 4: Draw a Graph for Lower Estimate

Draw a piecewise constant function where each interval from [0,5], [5,10], and [10,15] uses the value at the beginning of the interval. This will look like step-like bars of height 12, 20, and 24 respectively for each 5 minutes.

Key Concepts

Lower Estimate

When calculating the lower estimate for the volume of water that enters the rowboat, we use a method called the "left Riemann sum." This sum involves estimating the area under a curve by using rectangles aligned with left endpoints. Specifically, for each time interval, we take the measured inflow rate at the start of that interval.

• For the interval from 0 to 5 minutes, we use the rate at 0 minutes: 12 liters/min.
• For the interval from 5 to 10 minutes, we use the rate at 5 minutes: 20 liters/min.
• For the interval from 10 to 15 minutes, we use the rate at 10 minutes: 24 liters/min.

The lower estimate is then calculated by multiplying each rate by the duration of the interval (5 minutes), and summing these values:\[(5 \times 12) + (5 \times 20) + (5 \times 24) = 60 + 100 + 120 = 280 \text{ liters}\] This estimate gives us a baseline assumption, ensuring that we only consider minimum possible water accumulated given the rates.

Upper Estimate

To find the upper estimate, we use the "right Riemann sum" method. In this case, instead of using the rate at the start of each interval, we use the rate at the end. This method often gives a larger estimate than the lower one, as it assumes the maximum rate within the interval.

• For the interval from 0 to 5 minutes, we use the rate at 5 minutes: 20 liters/min.
• For the interval from 5 to 10 minutes, we use the rate at 10 minutes: 24 liters/min.
• For the interval from 10 to 15 minutes, we use the rate at 15 minutes: 16 liters/min.

The upper estimate is then calculated similarly by multiplying each rate by the interval duration and summing these values:\[(5 \times 20) + (5 \times 24) + (5 \times 16) = 100 + 120 + 80 = 300 \text{ liters}\] This estimate takes into account the highest possible accumulation of water, providing a safety margin for the actual situation.

Piecewise Constant Function

The concept of a "piecewise constant function" is crucial when visually representing Riemann sums on a graph. This type of function is made up of sections, or "pieces," where each section has a constant value over its specific interval. In the context of lower and upper estimates, it helps in visualizing how water flows into the boat over time.
When creating the graph:

• For the lower estimate, each rectangle correlates to the initial rate recorded at the start of each interval, creating ascending steps.
• For the upper estimate, the rectangles correspond to the rate at the end of each interval, often creating a higher step-like pattern.

The graph would look like a set of steps, with their heights determined by the water rate values, forming a simple and clear visual representation. Understanding this helps in comprehending how different rates affect the total water volume and why estimates vary.