# Actually Applicable Application Problems and Brainteasers/Rain Accumulation

## Overview

Students' task with these graphs is to estimate accumulation by sketching rectangles, multiplying to find each rectangle's area, and adding the rectangles' area to find the total area under the curve. (Technically this is integral calculus, though at a level to be accessible even to elementary students.)

This is an application, not just a brainteaser. The total amount of rain that has fallen or is expected to fall can be important for real world purposes like water conservation and flood safety.

## General Method

1. Mark the endpoints of the interval of interest.
2. Split the interval into equal subintervals. (How many? The problem may say or you may decide.)
3. Going up from each interval, draw a rectangle whose height is given by a point on the graph. (Should you use the left endpoint, right endpoint, or midpoint or the subinterval? The problem may say or you may decide.)
• The problem statement may use abbreviations for the number of rectangles and which point to use for the rectangle's height.
• For example, L6 means you should use left endpoints of six rectangles, M3 means you should use midpoints of three rectangles, and R8 means you should use right endpoints of eight rectangles.
4. Multiply to find each rectangle's area
• If you know how to calculate the area of a trapezoid, you may also use a trapezoid where one side goes between the two endpoints (so is usually slanted) and the other three sides are like in the rectangles.
• For example, T4 would mean to find the total area of four trapezoids.
5. Add to find the total area of the rectangles. This is an estimate for the total accumulation of rain.