The area of a spherical balloon expand at a rate of 0.4cm square/s .find the rate at which its volume expand when the radius is 12cm
a = 4πr^2
da/dt = 8πr dr/dt
v = 4/3 πr^3 = ar/3
dv/dt = 1/3 (r da/dt + a dr/dt)
Now just plug in your numbers
To find the rate at which the volume of the spherical balloon expands, we can use the formula for the volume of a sphere:
V = (4/3)πr^3
Where V is the volume and r is the radius of the sphere.
Differentiating both sides of the equation with respect to time (t) gives us:
dV/dt = (4/3)π(3r^2)(dr/dt)
This equation allows us to calculate the rate of change of volume with respect to time (dV/dt) by substituting the values for r and dr/dt into the equation.
Given that the radius (r) is 12 cm, we can substitute this value into the equation:
dV/dt = (4/3)π(3(12)^2)(dr/dt)
Next, we need to find the value of dr/dt, which represents the rate at which the radius is changing. This information is not given in the question. Assuming the balloon is expanding uniformly, we can assume that the rate at which the radius is changing is equal to the rate at which the surface area is increasing.
To find the rate at which the surface area of the balloon expands, we can use the formula for the surface area of a sphere:
A = 4πr^2
Differentiating both sides of the equation with respect to time (t) gives us:
dA/dt = 8πr(dr/dt)
Substituting the given rate of change of the surface area (dA/dt = 0.4 cm^2/s) and the radius (r = 12 cm) into the equation, we can solve for dr/dt:
0.4 = 8π(12)(dr/dt)
Simplifying and solving for dr/dt gives:
dr/dt = 0.4 / (8π(12))
Once we have the value of dr/dt, we can substitute it back into the equation for dV/dt:
dV/dt = (4/3)π(3(12)^2)(0.4 / (8π(12)))
Simplifying the equation, the value of dV/dt is:
dV/dt = 24 cm^3/s
Therefore, the volume of the balloon is expanding at a rate of 24 cm^3/s when the radius is 12 cm.