Respuesta :
Answer:
22670.8 cm³/min
Step-by-step explanation:
Given:
Radius of the balloon at a certain time (r) = 19 cm
Rate of growth of radius is, [tex]\frac{dr}{dt}=5\ cm/min[/tex]
The rate at which the air is pumped in the balloon can be calculated by finding the rate of increase in the volume of the balloon.
So, first we find the volume of the sphere in terms of 'r'. As the balloon is spherical in shape, the volume of the balloon is equal to the volume of a sphere. Therefore,
Volume of balloon is given as:
[tex]V=\frac{4}{3}\pi r^3[/tex]
Now, rate of increase of volume is obtained by differentiating both sides of the equation with respect to time 't'.
Differentiating both sides with respect to time 't', we get:
[tex]\frac{dV}{dt}=\frac{d}{dt}(\frac{4}{3}\pi r^3)\\\\\frac{dV}{dt}=\frac{4\pi}{3}(3r^2)(\frac{dr}{dt})\\\\\frac{dV}{dt}=4\pi r^2(\frac{dr}{dt})[/tex]
Now, plug in 19 cm for 'r', 5 cm per minute for [tex]\frac{dr}{dt}[/tex] and solve for [tex]\frac{dV}{dt}[/tex]. This gives,
[tex]\frac{dV}{dt}=4\pi (19 cm)^2(5\ cm/min)\\\\\frac{dV}{dt}=4\times 3.14\times 361\times 5\ cm^3/min\\\\\frac{dV}{dt}=22670.8\ cm^3/min[/tex]
Therefore, the rate at which the air is being pumped into the balloon is 22670.8 cm³/min.
