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.
dr/dt = (k T)
v= (4/3) pi r^3
dv/dt = (4 pi) r^2 dr/dt ( in other words proportional to surface area of sphere)
dv/dt = (4 pi) r^2 dr/dt = (4 pi k T ) r^2
To find the expression for the rate of change of the volume of a glass sphere in terms of the radius and temperature, we need to use the formula for the volume of a sphere. The volume of a sphere is given by the formula:
V = (4/3)πr³
where V is the volume and r is the radius of the sphere.
Now, we are given that the radius of the glass sphere is increasing over time in direct proportion to the temperature. Therefore, we can write:
r = kT
where k is a constant of proportionality.
To find the rate of change of the volume with respect to time, we need to differentiate the volume formula with respect to time (t). Let's denote dV/dt as the rate of change of the volume.
Differentiating both sides of the volume formula with respect to time, we get:
dV/dt = d/dt (4/3)πr³
Using the chain rule of differentiation, we can write:
dV/dt = (4/3)π * d(r³)/dt
Now, let's differentiate r³ with respect to t. We can write:
d(r³)/dt = 3r² * dr/dt
Substituting this back into our expression for dV/dt, we get:
dV/dt = (4/3)π * 3r² * dr/dt
Simplifying this expression, we get:
dV/dt = 4πr² * dr/dt
Now, we can substitute the value of r from the given relation r = kT into the above equation to get the final expression:
dV/dt = 4π(kT)² * (dr/dt)
So, the expression for the rate of change of the volume of the glass sphere in terms of the radius and temperature is given by:
dV/dt = 4π(kT)² * (dr/dt)
To find the expression for the rate of change of the volume of a glass sphere (V cm^3) with respect to time (t seconds), we can use the chain rule of differentiation.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.
Since the radius (r) increases with time (t) in direct proportion to the temperature (T) of the glass, we can express the relationship as r = kT, where k is a constant of proportionality.
Differentiating both sides of the equation r = kT with respect to time, we get:
dr/dt = k dT/dt.
Since dT/dt represents the rate of change of temperature with respect to time, we can assume it to be a constant and denote it as C.
So, we have dr/dt = C.
Substituting the value of r = kT in the formula for the volume, we get:
V = (4/3)π(kT)^3.
Differentiating both sides of the equation with respect to time, we can use the chain rule. The chain rule states that if y = f(u) and u = g(x), then dy/dx = f'(u) * g'(x), where f' denotes the derivative of f with respect to u, and g' denotes the derivative of g with respect to x.
Using the chain rule, we get:
dV/dt = (4/3)π * 3(kT)^2 * d(kT)/dt.
Since d(kT)/dt = C (as mentioned earlier), we can substitute it in the equation to get:
dV/dt = (4/3)π * 3(kT)^2 * C.
Simplifying further, we get the expression:
dV/dt = 4πC(kT)^2.
Therefore, the expression for the rate of change of the volume of the glass sphere (V cm^3) with respect to time (t seconds) is dV/dt = 4πC(kT)^2.