Mathematical Modeling: Problem Solving

Mathematical modeling involves using mathematical concepts to represent and solve real-world problems.

One common example is modeling motion using the relationship between speed, time, and distance.

Consider a motorboat that travels at 25 kilometers per hour in still water. It takes 20 minutes or one-third of an hour to go upstream and 15 minutes or one-fourth of an hour to return downstream. The distance in both directions remains the same. What is the speed of the current?

The river’s flow alters the boat’s effective speed—reducing it upstream and increasing it downstream.

Let a variable represent the speed of the current.

Upstream, the effective speed is 25 kilometers per hour minus the speed of the current. Downstream, it becomes 25 kilometers per hour plus the speed of the current.

The upstream distance is given by effective speed multiplied by one-third of an hour; downstream, it's multiplied by one-fourth.

Since the distances are equal, the product of speed and time for each trip must also be equal.

Solving this equation gives the current’s speed as approximately 3.57 kilometers per hour.