How fast does the radius of a spherical soap bubble change when you blow air into it at the rate of 15 cubic centimeters per second? Our known rate is dv/dt , the change in volume with respect to time, which is 15 cubic centimeters per second. The rate we want to find is dr/dt , the change in the radius with respect to time. Remember that the volume of a sphere is v=4/3Pir^3.
V= k r^3 you know k
dV/dt= 3kr^2 dr/dt
dr/dt=(dV/dt)/3k * r^2
Ohhhh okay thanks so much.. your a ship builder??? that's pretty awesome. That's a very interesting job :)
You are welcome. Actually I am retired now :)
To find the rate of change of the radius with respect to time (dr/dt), we can use the given information about the rate of change of volume with respect to time (dv/dt).
We know that the volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius of the sphere.
We can differentiate both sides of this equation with respect to time (t) to get:
dV/dt = d/dt [(4/3)πr^3]
The left-hand side represents the rate of change of the volume with respect to time (dv/dt), which is given as 15 cubic centimeters per second. So, we have:
15 = d/dt [(4/3)πr^3]
To solve for dr/dt, the rate of change of the radius with respect to time, we need to isolate dr/dt on one side of the equation. Let's break down the steps:
1. Multiply both sides of the equation by dt to remove it from the denominator:
15dt = d[(4/3)πr^3]
2. Integrate both sides of the equation with respect to t:
∫15dt = ∫d[(4/3)πr^3]
The left side becomes:
15t + C1
On the right side, we integrate the derivative of (4/3)πr^3 with respect to r:
(4/3)πr^3 + C2
Where C1 and C2 are constants of integration.
3. Rearrange the equation by subtracting (4/3)πr^3 from both sides:
15t + C1 - (4/3)πr^3 = C2
We can combine C1 and C2 into a single constant of integration, C.
4. Finally, to find dr/dt, the rate of change of the radius with respect to time, we isolate it on one side of the equation:
dr/dt = (15t + C) / (4πr^2)
Therefore, the rate of change of the radius with respect to time, dr/dt, is given by (15t + C) / (4πr^2), where t is the time in seconds and r is the radius of the sphere.
The change of volume of a sphere with increasing radius is the surface area of the sphere times dr.
Draw a picture !
In other words:
dV = 4 pi r^2 dr
so
dV/dt = 4 pi r^2 dr/dt
so here
15 = 4 pi r^2 dr/dt
or
dr/dt = 15 /(4 pi r^2)
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note , I am a ship builder so usually do this sort of problem this way.
calculus experts would not draw a picture but take the derivative of the volume:
V = (4/3) pi r^3
so
dV/dr = 4 pi r^2 (which we knew)
so
dV/dt = dV/dr*dr/dt = 4 pi r^2 dr/dt
which we also knew :)