Question

can someone help me please!!! its calc 1 related rates! im so confused on the whole concept.
ex- air is being pumped into a spherical ballon so that its volume increases at a rate of 100cm^3/s. how fast is the radius of the ballon increasing when the diameter is 50cm?

Answer 1

It revolves around using implicit differentiation and remembering the relationship for the volume of a sphere.
\[V=\frac{4\pi}{3}r^3\]
\[\frac{dV}{dt}=100=\frac{dV}{dr}\frac{dr}{dt}\]
We know r=25, so we can find
\[\frac{dV}{dr}=4\pi r^2\] and thus solve for \[\frac{dr}{dt}=\frac{dV/dt}{dV/dr}=\frac{100}{4\pi*25^2}\] which I am too lazy to calculate.

Answer 2

Clues are question asks for how "fast" so we know we need to think of radius changing with time, and fact given the Rate the volume is changing, so know need to relate volume to radius.