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Related Rates Visualizer
AP Calculus AB · Unit 4 Contextual Applications of Differentiation
Learning Objectives
With Respect to Time (t):
ALL variables (V, r, h, A) are changing functions of time. You MUST differentiate with respect to $dt$.
Chain Rule Injection:
When taking the derivative of $r^2$ with respect to $t$, the result is $2r \cdot \frac{dr}{dt}$.
The Paradox of Rates:
Even if volume increases at a CONSTANT constant rate, the radius increases slower and slower as the object gets larger.
Tags
Derivatives
Related Rates
Geometry
Time (t) = 0.0s
Select Scenario
Spherical Balloon
$V = \frac{4}{3}\pi r^3$
Draining Cone
$V = \frac{1}{3}\pi r^2 h$
Play Simulation
Instantaneous Rates of Change
CONSTANT INFLATION RATE
Pump Rate $\frac{dV}{dt}$
100 cm³/s
Radius $r$
0.00 cm
DERIVATIVE EQUATION
$\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt}$
$\frac{dr}{dt} = $
0.00
CONSTANT DRAIN RATE
Drain Rate $\frac{dV}{dt}$
-50 cm³/s
Water Level $h$
0.00 cm
Geometry Ratio
$r = \frac{h}{2}$
DERIVATIVE EQUATION
$\frac{dV}{dt} = \frac{\pi}{4} h^2 \cdot \frac{dh}{dt}$
$\frac{dh}{dt} = $
0.00
Spherical Balloon Paradox:
Watch the $\frac{dr}{dt}$ value. Because the balloon's surface area ($4\pi r^2$) grows drastically as it inflates, a constant supply of air ($\frac{dV}{dt} = 100$) causes the radius to grow SLOWER AND SLOWER.