Applications of derevatives

Question

A gas is escaping from spherical balloon at rate of 900cm^3/sec.How fast surface area shrinking when radius is 360cm

Answer 1

Lets call the radius of the balloon r. We know from the problem that the balloon is spherical so we can use the formula for the volume V of a sphere which is

V = 4/3 * π * r³

Now taking the derivative of the volume will give us a formula for how fast the balloon is shrinking (note: r is radius). From the problem, they tell us that the balloon is shrinking 900cm^3/sec, and since the rate of shrinking is the derivative of volume, dV/dt = 900cm^3/sec

So dV/dt = 4*π*r² * dr/dt = 900cm^3/sec

From this equation, we solve for dr/dt, how fast the radius is shrinking.

dr/dt = 900 / (4*π*r² )

Now we want to think about the surface area. We can use the surface area formula of a sphere:

S = 4*π*r²

To find how the surface area is shrinking, we again need to take a derivative, this time the derivative of the surface area.

dS/dt = 8 * π * r * dr/dt

Now we plug in the value of dr/dt that we solved for above.

dS/dt = 8*π*r *900 / (4*π*r² ) = 2*900/r

The question asks how fast the surface area is shrinking (what is dS/dt) when the radius is 360 (r=360). So to get the final answer, we just need to plug in r=360.

dS/dt = 2*900/360 = 1800/360 = 5.

Hope this helps, let me know if you have more questions about this question or others!