Respuesta :
Using implicit differentiation, it is found that she must blow air at a rate of 251.33 cm³/sec.
- A balloon has a spherical format.
The volume of a sphere of radius r is given by:
[tex]V = \frac{4\pi r^3}{3}[/tex]
The radius is half the diameter, hence:
[tex]r = \frac{d}{2}[/tex]
[tex]r^3 = \frac{d^3}{8}[/tex]
[tex]V = \frac{4\pi d^3}{24} = \frac{\pi d^3}{6}[/tex]
Applying implicit differentiation, the rate of change of the volume is:
[tex]\frac{dV}{dt} = \frac{d^2 \pi}{2}\frac{dd}{dt}[/tex]
- The diameter increases at the rate of 10 cm/sec, hence [tex]\frac{dd}{dt} = 10[/tex].
- It is of 4 cm, hence [tex]d = 4[/tex]
The rate that she must blow air is:
[tex]\frac{dV}{dt} = \frac{d^2 \pi}{2}\frac{dd}{dt}[/tex]
[tex]\frac{dV}{dt} = \frac{4^2 \pi}{2}(10)[/tex]
[tex]\frac{dV}{dt} = 251.33[/tex]
She must blow air at a rate of 251.33 cm³/sec.
A similar problem is given at https://brainly.com/question/25581404
