Related Rates:
Gas is escaping from a spherical balloon at the rate of 2ft^3/min. How fast is the surface area shrinking (ds/dt) when the radius is 12ft? (A sphere of radius r has volume v=4/3 pi r^3 and surface area S=4pi r^2.)
Remember that ds/dt = ds/dr X dr/dt
Step 1: Find ds/dr
I have no clue where to start. I was going to set 2 equal to 4 pi r^2 and then take the derivative but I really have no idea.
Step 2: Find dr/dr. (HInt dv/dt= dv/dr X dr/dt
Would I set 2 equal to volume and then take derivative?
Step 3: Find ds/dt?
Step 4: Evaluate ds/dt when the radius is 12ft.
After I find my equation I would just plug in 12 correct?
S = 4pi*r^2
dS/dr = 4pi*2r = 8pi*r
V = (4/3)pi*r^3
dV/dr = 4pi*r^2
dV/dt = dV/dr * dr/dt
2 = 4pi*r^2 * dr/dt
dr/dt = 1/(2pi*r^2)
dS/dt = dS/dr * dr/dt
= 8pi*r * 1/(2pi*r^2)
= 4/r
If r = 12, just plug the value to the last equation
Ok do I actually have to do the calculations though?
Simplify the answer (4/12) into 1/3
To solve this related rates problem, let's go step by step.
Step 1: Find ds/dr (the rate at which the surface area is changing with respect to the radius).
We know that the surface area of a sphere is given by S = 4πr^2. To find ds/dr, we need to take the derivative of S with respect to r.
So, differentiate S = 4πr^2 with respect to r:
dS/dr = d(4πr^2)/dr = 8πr
Therefore, ds/dr = 8πr.
Step 2: Find dr/dt (the rate at which the radius is changing with respect to time).
Given that the volume of a sphere is V = (4/3)πr^3, we can find dr/dt using the relationship dv/dt = (dv/dr) * (dr/dt).
The rate at which the volume is changing, dv/dt, is given as 2 ft^3/min.
Differentiate V = (4/3)πr^3 with respect to r:
dV/dr = d((4/3)πr^3)/dr = 4πr^2
Now, use dv/dt = (dv/dr) * (dr/dt) and substitute values:
2 = (4πr^2) * (dr/dt)
Solving for dr/dt:
dr/dt = 2 / (4πr^2)
dr/dt = 1 / (2πr^2)
Step 3: Find ds/dt (the rate at which the surface area is changing with respect to time).
We can now find ds/dt using the formula ds/dt = (ds/dr) * (dr/dt). Substitute the values we found in Steps 1 and 2.
ds/dt = (8πr) * (1 / (2πr^2))
ds/dt = 4 / r
Step 4: Evaluate ds/dt when the radius is 12 ft.
To evaluate ds/dt when the radius is 12 ft, substitute r = 12 into the equation we found in Step 3.
ds/dt = 4 / 12
ds/dt = 1/3 ft^2 per minute
Therefore, the surface area is shrinking at a rate of 1/3 square feet per minute when the radius is 12 ft.
If you are not given the value, that means you have to write the steps I told you until the very last equation.
If you are given the value (of r), you don't have to put the value in every equation. Just follow my steps until the last equation, then plug the value after that. It's easier that way.
OK thank you!
Is this correct for the last part?
(8)(3.14)(12) X (1/(2)(3.14)(12^2) = 4/12