A snowball has a radius of 3 inches. Assume the rate with which the volume of the snowball melts is proportional to its surface area. If, after 1 hour, the radius of the snowball is 2.9 inches, predict what the radius will be after one day.
dv/dt = k (4 pi r^2) given
v = (4/3) pi r^3 geometry
dv/dr = 4 pi r^2
dv/dt = dv/dr dr/dt = 4 pi r^2 dr/dt
so
k is dr/dt which is constant
dr/dt = (2.9 - 3)/1 = -0.1 in/hr
starts at 3 in
r = 3 - .1(24) = 3-2.4 = 0.6 inches
can you solve it with separation of differential equation
To solve this problem, we need to use the formula for the surface area and volume of a sphere.
First, let's calculate the initial surface area and volume of the snowball when the radius is 3 inches.
The formula for the surface area of a sphere is given by:
Surface Area = 4πr^2
Plugging in the radius of 3 inches:
Surface Area = 4π(3^2) = 4π(9) = 36π square inches
The formula for the volume of a sphere is given by:
Volume = 4/3πr^3
Plugging in the radius of 3 inches:
Volume = 4/3π(3^3) = 4/3π(27) = 36π cubic inches
Now, let's calculate the surface area and volume of the snowball after 1 hour when the radius is 2.9 inches.
Surface Area = 4π(2.9^2) = 4π(8.41) ≈ 33.64π square inches
Volume = 4/3π(2.9^3) ≈ 4/3π(24.389) ≈ 102.364π cubic inches
We are given that the rate at which the volume of the snowball melts is proportional to its surface area. Therefore, we can set up the following proportion:
(initial surface area - surface area after 1 hour) / (initial volume - volume after 1 hour) = (initial radius - radius after 1 day) / (initial radius - radius after 1 hour)
Plugging in the values we calculated:
(36π - 33.64π) / (36π - 102.364π) = (3 - radius after 1 day) / (3 - 2.9)
2.36π / (36π - 102.364π) = 0.1 / (3 - 2.9)
Simplifying the equation further:
2.36π / -66.364π = 0.1 / 0.1
Dividing both sides by 0.1:
2.36π / -66.364π ≈ 1
Simplifying the equation further:
2.36π = -66.364π
Dividing both sides by π:
2.36 ≈ -66.364
Since this equation is not possible, we have made an error in our calculations or assumptions.
Please check the given information and calculations for accuracy.