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.
The question seems kinda murky. If you mean
dr/dt = kT
v = 4/3 πr^3
dv/dt = 4πr^2 dr/rt = 4π(kTt)^3
"y is directly proportional to x" means that
y = kx
for some constant k.
C'mon, man, that is Algebra I
To find the expression for the rate of change of the volume of a glass sphere, we need to use the formula for the volume of a sphere, which is:
V = (4/3) * π * r^3
Where V represents the volume and r represents the radius of the sphere.
Since the radius is increasing over time in proportion to the temperature, we can write the rate of change of the radius as:
dr/dt = k * T
Where dr/dt represents the rate of change of the radius, k is the proportionality constant, and T is the temperature.
Now, we can differentiate the volume formula with respect to time (t) using the chain rule:
dV/dt = dV/dr * dr/dt
To find dV/dr, we can differentiate the volume formula with respect to r:
dV/dr = (4/3) * π * 3 * r^2
dV/dr simplifies to:
dV/dr = 4πr^2
Now, substitute the value of dV/dr into the equation for dV/dt:
dV/dt = (4πr^2) * (kT)
Therefore, the expression for the rate of change of the volume (dV/dt) in terms of the radius (r) and temperature (T) is:
dV/dt = 4πr^2 * kT