Problem 69 - Calculus for Biology and Medicine

Question

Total Rainfall A rain gauge is set up to measure the amount of rainfall occurring in \(1 \mathrm{hr}\) on the UCLA campus (the readout from the rain gauge is in \(\mathrm{mm} / \mathrm{hr}\) ). Assume that the following data is collected in a 6 hour window. $$ \begin{array}{c|c} \hline \text { Time, } t & \text { Rainfall rate, } \boldsymbol{r}(\boldsymbol{t}) \text { in } \mathrm{mm} / \mathrm{hr} \\ \hline 0 & 1 \\ 1 & 2 \\ 2 & 3 \\ 3 & 1 \\ 4 & 1 \\ 5 & 0 \\ 6 & 0 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The total rainfall over the 6-hour period is 8 mm.

Step 1: Understanding the Problem

We need to calculate the total rainfall over a 6-hour period based on the given data, where each time interval is 1 hour, and rainfall rates are in mm/hr.

Step 2: Calculation of Total Rainfall

To find the total rainfall over the 6-hour period, we compute the sum of the rainfall rates for each hour. This is given by the formula: Total Rainfall = \( r(0) + r(1) + r(2) + r(3) + r(4) + r(5) + r(6) \). Substitute the respective rates: \( 1 + 2 + 3 + 1 + 1 + 0 + 0 \).

Step 3: Sum the Rainfall Rates

Add the rainfall rates based on the time intervals provided: \( 1 + 2 + 3 + 1 + 1 + 0 + 0 = 8 \).

Step 4: Interpret the Result

The sum of the rainfall rates gives us the total rainfall in millimeters over the 6-hour period. Therefore, the total rainfall is 8 mm.

Key Concepts

Integration

Integration is a key concept in calculus, often used to calculate areas under curves. It can be challenging at first but becomes manageable once you see its practical applications, like calculating total quantities over time. In the context of rainfall measurement, integration makes it possible to determine the overall rainfall during a specific timeframe by summing up instantaneous rates of rain.

When we talk about integrating the rainfall rate, it's about adding up small slices of rain that fall over each interval. If you imagine a graph where the x-axis represents time, and the y-axis indicates the rainfall rate, then the integral of this graph (area under the curve) gives the total rain accumulated during that period. This doesn't just apply to rain, but to any scenario where a total quantity is gathered over time or along a path.

• Integration involves summing up infinitesimally small quantities over a given interval.
• It is akin to finding the area under the curve in a graph of rate versus time.
• For discrete data, like in this exercise, summing the rates directly represents integration in practice.

Rainfall Measurement

Measuring rainfall accurately is crucial for various fields, especially in agriculture and meteorology. A rain gauge collects rain over time and provides data in terms of rainfall rate (e.g., mm/hr). Understanding these rates helps in analysis and prediction of weather patterns.

In the problem provided, the rain gauge measures rainfall at different time intervals, creating a data set of rates. These rates show how much rain falls each hour. To find out how much rain falls in total, we sum these rates over the period, which was effectively performed during the exercise.

• Rain gauges are tools for collecting rain and measuring the amount of rainfall over a set time.
• The data output, usually in mm/hr, represents how much rain is falling over each hour.
• Interpreting this data allows us to determine total rainfall over the period measured by calculating the sum of these rates.

Data Analysis

Analyzing rainfall data involves interpreting the collected measurements to make informed conclusions or predictions. This type of data analysis can provide insights into rainfall patterns, intensity, and behavior over time.

In our example, data is collected every hour, showing different rates. Through straightforward analysis, we simply sum these hourly intervals to find the total rainfall over 6 hours. Although this is basic, more advanced analyses might look for patterns over a season or compare data with previous years.

• Data analysis involves collecting and interpreting data to find patterns or make predictions.
• In our simple example, summing the hourly rain rates allows us to determine the total rainfall over a period.
• Advanced analysis can include comparing data across different timeframes or analyzing rainfall's impact on the environment.