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?

1. 👍
2. 👎
3. 👁
1. 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

1. 👍
2. 👎
2. How did you get 12?

1. 👍
2. 👎
3. Nevermind I meant how did you get 4/r but I figured it out. Thank You!!

1. 👍
2. 👎
4. Is this correct for the last part?

(8)(3.14)(12) X (1/(2)(3.14)(12^2) = 4/12

1. 👍
2. 👎
5. Yes

1. 👍
2. 👎
6. Ok do I actually have to do the calculations though?

1. 👍
2. 👎
7. 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.

1. 👍
2. 👎
8. OK thank you!

1. 👍
2. 👎
9. Simplify the answer (4/12) into 1/3

1. 👍
2. 👎

## Similar Questions

1. ### calculas

i didnot under stand the question ? need help Gas is escaping a spherical balloon at the rate of 4 cm3 per minute. How fast is the surface area of the balloon shrinking when the radius of the balloon is 24 cm? Given volume of

2. ### Math

A spherical balloon is to be deflated so that its radius decreases at a constant rate of 12 cm/min. At what rate must air be removed when the radius is 5 cm? Must be accurate to the 5th decimal place. It seems like an easy

3. ### Word Problem-Calculus

A spherical balloon is being inflated in such a way that its radius increases at a rate of 3 cm/min. If the volume of the balloon is 0 at time 0, at what rate is the volume increasing after 5 minutes? my answer is 45 cm/min. is

4. ### Calculus

a spherical balloon is inflated with gas at the rate of 500 cubic centimeters per minute. how fast is the radius of the balloon increasing at the instant the radius is 30 centimeters?

1. ### Calculus

A spherical balloon is inflated at a rate of 10 cubic feet per minute. How fast is the radius of the balloon changing at the instant the radius is 4 feet? And The radius of a circle is decreasing at a rate of 2 ft/minute. Find the

2. ### help on calculus

A spherical balloon is being filled with air in such a way that its radius is increasing at a rate of 2 centimeters per second. At what rate is the volume of the balloon changing at the instant when its surface has an area of 4 pi

3. ### Calculus

Air is being pumped into a spherical balloon so that its volume increases at a rate of 30 cubic centimeters per second. How fast is the surface area of the balloon increasing when its radius is 19 cm? V= 4/3*pi*r^3 S= 4 pi r^2

4. ### calculus help thanks!

The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nRT, where n is the number of moles of the gas and R = 0.0821 is the gas constant. Suppose that, at

1. ### Math

A spherical balloon is inflated with helium at the rate of 100pi ft ^3/min. How fast is the ballon's radius increasing at the instant the radius is 5ft.?

2. ### Mathematics

A spherical balloon is blown up so that it's volume increases constantly at the rate of 2cm^3 per second .find the rate of the increase of the radius when the volume of the balloon is 50cm^3

3. ### defferential calculus

A balloon is rising vertically 150 m from an observer. At exactly 1 min, the angle of elevation is 29 deg 28 min. How fast is the balloon rising at that instant?

4. ### Calculus

A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2cm/min. At what rates are the volume and surface area of the balloon increasing when the radius is 5cm?