In the production process of a glass sphere, hot glass is blown such that the radius, r cm, increases over time (t seconds) in direct proportion to the temperature (T °C) of the glass.
Find an expression, in terms of r and T, for the rate of change of the volume (V cm^3) of a glass sphere.
Well, well, well, looks like we have ourselves a hot and steamy glass sphere on our hands! Let's dive into the fiery world of rate of change, shall we?
To start things off, we can use the formula for the volume of a sphere: V = (4/3)πr³. Now we need to figure out how the volume changes with respect to time.
Since the radius is in direct proportion to the temperature, we can say that r = kT, where k is just some constant of proportionality. (I know, it sounds like a bad pick-up line for a glass sphere, but bear with me here.)
Now, let's differentiate both sides of this equation with respect to time (t). Using the chain rule, we get:
(d/dt) r = (d/dt) (kT)
The left side represents the rate of change of the radius, which we'll call dr/dt. And since the temperature (T) doesn't change with time, the right side becomes:
dr/dt = k (d/dt) T
But we're not looking for the rate of change of the radius - we want the rate of change of the volume, which is V. So let's differentiate the volume equation with respect to time (t):
(d/dt) V = (d/dt) [(4/3)πr³]
Using the chain rule again, we get:
(d/dt) V = (4/3)π (d/dt) (r³)
However, we know that dr/dt = k (d/dt) T from before. So we can rewrite the right side as:
(d/dt) V = (4/3)π (d/dt) (k³T³)
Applying the power rule and multiplying through, we get:
(d/dt) V = 4πk³ (d/dt) T³
And remember, (d/dt) T³ is just the rate of change of the temperature with respect to time. So, in the end, we have our expression for the rate of change of the volume in terms of r and T:
(d/dt) V = 4πk³ (d/dt) T³
Now you can go impress your friends with your knowledge of hot glass spheres and their changing volumes. Just make sure you don't get burned in the process!
To find the rate of change of the volume of a glass sphere with respect to time, we need to use the chain rule from calculus.
The volume (V) of a sphere is given by the formula: V = (4/3)πr^3, where r is the radius.
First, let's find the derivative of V with respect to r: dV/dr = 4πr^2.
Next, to find the rate of change of the volume with respect to time, we need to find the derivative of V with respect to t. To do this, we can use the chain rule.
Since the radius (r) is changing over time (t) and is in direct proportion to the temperature (T), we can express r as a function of T: r = kT, where k is a constant of proportionality.
Now we can differentiate the volume with respect to time using the chain rule:
dV/dt = (dV/dr) * (dr/dt)
Substituting the expressions we have, we get:
dV/dt = (4πr^2) * (dr/dt)
Since r = kT, we can substitute it back into the equation:
dV/dt = (4π(kT)^2) * (dr/dt)
Simplifying this expression gives us the final answer:
dV/dt = 4πk^2T^2 * (dr/dt)
So, the expression for the rate of change of the volume (V) of a glass sphere with respect to time (t) is 4πk^2T^2 * (dr/dt).
To find the expression for the rate of change of the volume of a glass sphere, we need to differentiate the volume function with respect to time.
First, let's determine the volume of a sphere in terms of its radius:
The formula for the volume of a sphere is V = (4/3) * π * r^3, where r is the radius of the sphere.
Now, we need to express the radius, r, in terms of time, t, and the temperature, T.
According to the problem, the radius increases over time in direct proportion to the temperature of the glass. Therefore, we can write:
r = k * T, where k is a constant of proportionality.
Now, let's substitute this value of r in terms of T into the volume equation:
V = (4/3) * π * (k * T)^3.
Simplifying, we get:
V = (4/3) * π * k^3 * T^3.
To find the rate of change of volume with respect to time, we differentiate the volume equation with respect to time, t:
dV/dt = d/dt [(4/3) * π * k^3 * T^3]
Using the power rule of differentiation, we can simplify this expression:
dV/dt = (4/3) * π * 3 * k^3 * T^2 * (dT/dt)
Simplifying further, we obtain:
dV/dt = 4 * π * k^3 * T^2 * (dT/dt)
Therefore, the expression for the rate of change of volume (dV/dt) in terms of radius (r) and temperature (T) is:
dV/dt = 4 * π * (r^3 / k^3) * T^2 * (dT/dt).