A spherical balloon is inflating with helium at a rate of 192pi ft^3/min. How fast is the balloon's radius increasing at the instant the radius is ft?

3ft/min

Explanation:

Complete question:

A spherical balloon is inflating with helium at a rate of 192pi ft^3/min. How fast is the balloon's radius increasing at the instant the radius is 4ft?

Volume of a sphere = where:

r is the radius of the sphere:

Using the chain rule to find dV/dt we have:

is the rate at which the balloon's radius is increasing

is the rate at which the volume is increasing

Given

radius = 4ft

Required

From the formula:

Get

Hence the balloon radius is increasing at the rate of 3ft/min.

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.