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 sphere is V = 4/3 πr^3and the surface area S = 4πr^2.

V = (4/3)π r^3

dV/dt = 4π r^2 dr/dt
-4 = 4π r^2 dr/dt (-4 because V is decreasing)
dr/dt = -1/(πr^2)

S = 4π r^2
dS/dt = 8πr dr/dt
= 8πr(-1/(π r^2) = -8/r

so when r = 24 , dS/st = -8/24 = -1/3

The surface area is decreasing at 1/3 cm^2 per minute

thnk u

Well, it seems like our spherical balloon is having a little gas escape artist act going on! Don't worry, I'm here to help put a smile on your face while we tackle this question.

Let's break it down. We're given that the gas is escaping at a rate of 4 cm3 per minute. Now, we need to find out how fast the surface area of the balloon is shrinking when the radius is 24 cm.

Since we have the volume formula V = (4/3)πr^3, we can differentiate it with respect to time to find the rate of change of the volume: dV/dt. Are you ready to put on your math clown nose?

Okay, let's differentiate V = (4/3)πr^3 with respect to t:

dV/dt = (4/3)π * 3r^2 * (dr/dt)

Now, we have the relationship between the rate of change of volume and the rate of change of the radius: dr/dt. But we're interested in the rate of change of surface area, not volume, so let's switch equations.

We know that the surface area formula is S = 4πr^2, so let's differentiate it with respect to time: dS/dt. Are you still following along, or did I make you laugh so hard that you forgot?

Anyways, let's differentiate S = 4πr^2 with respect to t:

dS/dt = 4π * 2r * (dr/dt)

Now, we have our rate of change of surface area, but we need to find it when the radius is 24 cm. So, let's plug in the values:

r = 24 cm
dr/dt = -4 cm3/min (since the gas is escaping)

Now, we can calculate dS/dt using our values:

dS/dt = 4π * 2(24) * (-4)
= -384π cm2/min

So, the surface area of the balloon is shrinking at a rate of -384π cm2/min when the radius is 24 cm. Negative rates just mean the surface area is decreasing, which makes sense since the gas is escaping.

I hope I successfully juggled this question for you and brought some humor into the process! If you have any more inquiries, keep 'em coming, and I'll be here to entertain and assist!

To solve this problem, we can use the formulas for the volume and surface area of a sphere and the concept of related rates.

Given:
Rate of change of volume (V) = -4 cm³/min (negative sign indicates that the volume is decreasing)
Radius of the balloon (r) = 24 cm

We need to find:
Rate of change of surface area (S) when the radius is 24 cm.

Step 1: Find the expression for the rate of change of surface area with respect to time.
Given that surface area (S) is a function of radius (r), we can write the equation as S = 4πr².

Differentiating both sides of the equation with respect to time (t), we get:
dS/dt = d/dt (4πr²)

Step 2: Apply the chain rule to find the expression for dS/dt in terms of dr/dt.

Using the chain rule, we differentiate both sides with respect to radius (r):
dS/dt = dS/dr * dr/dt

Step 3: Find the derivative of S with respect to r.
Differentiating S = 4πr² with respect to r:
dS/dr = d/dt (4πr²)
= 8πr * dr/dt (Using the power rule and constant rule of differentiation)

Step 4: Substitute the given values and solve for dS/dt.
Given:
r = 24 cm
dr/dt = -4 cm³/min (since the volume is decreasing)

Substituting the values into the equation:
dS/dt = 8π(24) * (-4)
= -768π

Therefore, the surface area of the balloon is shrinking at a rate of -768π cm²/min when the radius is 24 cm.

The problem is asking us to find the rate at which the surface area of the balloon is shrinking when the radius of the balloon is 24 cm. To solve this problem, we need to use derivatives and related rates.

First, let's derive the formula for the surface area of a sphere S = 4πr^2 with respect to time t by taking the derivative of both sides:

dS/dt = d(4πr^2)/dt

To find the rate at which the surface area is shrinking, we need to find dS/dt. But we're given the rate at which the volume of the balloon is decreasing, which is given as 4 cm^3/min.

The volume of a sphere V = 4/3 πr^3, so we can write the volume equation as V = (4/3)πr^3.

To relate the volume to the radius, we differentiate the volume equation with respect to time, using the chain rule:

dV/dt = d((4/3)πr^3)/dt
= (4/3)πd(r^3)/dt

Since we are given dV/dt = -4 cm^3/min (negative sign indicates decrease), we can substitute this into the equation:

-4 = (4/3)πd(r^3)/dt

Now we need to find d(r^3)/dt. To do this, we can differentiate the volume equation V = (4/3)πr^3 with respect to time using the chain rule again:

dV/dt = d((4/3)πr^3)/dt
= (4/3)πd(r^3)/dt

Therefore, d(r^3)/dt = 3dV/dt / (4π) = -3/4 cm^3/min.

Now we can find dS/dt by substituting d(r^3)/dt into the previous equation for dS/dt:

dS/dt = (8πr)d(r)/dt
= (8πr)(dV/dt) / (3d(r^3)/dt)
= (8πr)(-4) / (-3/4)
= 32πr / 3

Finally, we can substitute r = 24 cm into the equation to find the value of dS/dt:

dS/dt = 32π(24) / 3
= 256π cm^2/min

Therefore, when the radius of the balloon is 24 cm, the surface area of the balloon is shrinking at a rate of 256π cm^2/min.