Im trying to understand related rates here and im getting stuck at this word problem.
It says if V is the volume of a sphere with radius r, and the volume of the sphere decreases as the time passes. express dV/dt in terms of dr/dt. Describe the relationship in words.
I don't get what it means by dV/dt in terms of dr/dt, i know what the derivatives and related rates mean, but I have trouble connecting them and saying it in words.
V = (4/3) pi r^3
dV/dt = 4 pi r^2 dr/dt
Put that into words
To understand the relationship between dV/dt and dr/dt, let's break down the problem step-by-step and explain each component.
First, let's consider the formula for the volume of a sphere: V = (4/3) π r^3. Here, V represents the volume, and r represents the radius of the sphere.
Now, we are interested in how the volume changes over time, or in other words, the derivative of V with respect to time, which is represented by dV/dt. This tells us how fast the volume of the sphere is changing.
To express dV/dt in terms of dr/dt, we need to find a relationship between the rate at which the volume changes (dV/dt) and the rate at which the radius changes (dr/dt).
To do this, we can differentiate the volume formula with respect to time using the chain rule. The derivative of V with respect to time (dV/dt) is equal to the derivative of V with respect to r (dV/dr) multiplied by the derivative of r with respect to time (dr/dt).
The derivative of V with respect to r is found by differentiating the volume formula with respect to r, which gives us dV/dr = 4πr^2. This represents the rate at which the volume changes concerning the radius.
Finally, we substitute dV/dr = 4πr^2 and dr/dt into the equation to find the relationship between dV/dt and dr/dt:
dV/dt = (4πr^2) * (dr/dt)
This equation shows that the rate of change of the volume (dV/dt) is equal to the rate of change of the radius (dr/dt) multiplied by a constant value of 4πr^2.
In words, the relationship can be described as follows: "The rate at which the volume of the sphere is changing (dV/dt) equals the rate at which the radius is changing (dr/dt) multiplied by 4πr^2." This means that as the radius changes, the volume changes at a rate proportional to the current radius multiplied by a constant value of 4πr^2.