Related Rates Triangle Calculator
Results
Hypotenuse Length (c): 0
Rate of Change (dc/dt): 0 units/time
Calculation based on 2a(da/dt) + 2b(db/dt) = 2c(dc/dt)
Understanding Related Rates in Right Triangles
Related rates problems are a core component of differential calculus. They involve finding the rate at which one quantity changes by relating that quantity to others whose rates of change are already known. In the context of a right triangle, we typically use the Pythagorean Theorem as our primary constraint.
The Mathematical Foundation
For a right triangle with legs a and b and hypotenuse c, the relationship is defined by:
a² + b² = c²
To find the related rates, we differentiate both sides with respect to time (t):
2a(da/dt) + 2b(db/dt) = 2c(dc/dt)
By simplifying (dividing by 2), we get the formula used by this calculator:
dc/dt = [a(da/dt) + b(db/dt)] / c
Real-World Example: Two Cars at an Intersection
Imagine Car A is traveling East away from an intersection at 30 mph, and Car B is traveling North away from the intersection at 40 mph. At the moment Car A is 3 miles from the intersection and Car B is 4 miles from the intersection, how fast is the distance between them increasing?
- Side a: 3 miles
- da/dt: 30 mph
- Side b: 4 miles
- db/dt: 40 mph
First, find the distance c using the Pythagorean Theorem: 3² + 4² = c², so 9 + 16 = 25, meaning c = 5.
Now, plug the values into the rate formula: (3 * 30 + 4 * 40) / 5 = (90 + 160) / 5 = 250 / 5 = 50 mph. The distance between the cars is increasing at 50 mph.
How to Use This Calculator
1. Input Lengths: Enter the current lengths of the two legs (a and b) of the triangle.
2. Input Rates: Enter how fast those lengths are changing. Use positive values for increasing lengths and negative values for decreasing lengths (e.g., if a ladder is sliding down a wall, its height rate would be negative).
3. Interpret Results: The calculator provides the current length of the hypotenuse and the instantaneous rate at which the hypotenuse is changing.