A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 16 cm.
The diameter of the sphere is decreasing at 0.2 cm/min. This means that its radius is decreasing by 0.1 cm/min.
The formul
The diameter of the sphere is decreasing at 0.2 cm/min. This means that its radius is decreasing by 0.1 cm/min. We also know its current radius is 8 cm.
The formula for the volume of a sphere is:
V= (4/3)*Pi*r^3
So, we can calculate the volume difference in 1 minute, by subtracting its volume after one minute from its current volume:
Vc-Vf = [(4/3)*Pi*(8 cm)^3] -
[(4/3)*Pi*(8-0.1 cm)^3]
= 2144.66 cm^3- 2065.24 cm^3= 79.42 cm^3
This means its volume is decreasing by 79.42 cm^3/min
Hmm. I tried that and it wasn't correct.
The answer is 80.4248 cm^3/min.
(1/3)(4π)(1/8)(3(16)²)(-0.2)
To find the rate at which the volume of the snowball is decreasing, 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. Since we are given the rate at which the diameter is decreasing, we can find the rate at which the radius is decreasing using the relationship between diameter and radius:
r = d/2
Where r is the radius and d is the diameter.
Differentiating both sides of this equation with respect to time, we get:
dr/dt = (1/2) * dd/dt
Where dr/dt is the rate at which the radius is changing and dd/dt is the rate at which the diameter is changing.
Given that dd/dt = -0.2 cm/min (since the diameter is decreasing), we can calculate dr/dt:
dr/dt = (1/2) * (-0.2 cm/min) = -0.1 cm/min
Now we have the rate at which the radius is decreasing. To find the rate at which the volume is decreasing, we differentiate both sides of the volume formula with respect to time:
dV/dt = 4πr^2 * dr/dt
Substituting the values we have:
dV/dt = 4π(16 cm)^2 * (-0.1 cm/min) = -804.25π cm^3/min
Therefore, the volume of the snowball is decreasing at a rate of approximately -804.25π cm^3/min when the diameter is 16 cm.