Respuesta :
Answer:
Explanation:
[tex]\frac{dr}{dt}[/tex] = 1.6in/s, [tex]\frac{dh}{dt}[/tex] = -2.4 in/s radius = 129 in and height = 128 in
the volume of a right circular cone = [tex]\frac{1}{3} \pi r^{2} h[/tex]
using chain rule equation to determine the rate of change in volume
[tex]\frac{dv}{dt}[/tex] = [tex]\frac{dv}{dr}[/tex] ([tex]\frac{dr}{dt}[/tex]) + [tex]\frac{dv}{dh}[/tex]( [tex]\frac{dh}{dt}[/tex])
partial differentiating with respect to radius and height respectively
[tex]\frac{dv}{dr}[/tex] = [tex]\frac{d}{dr}[/tex]( [tex]\frac{1}{3}\pi r^{2} h[/tex]) = [tex]\frac{2}{3}\pi rh[/tex] = 11008 π
[tex]\frac{dv}{dh}[/tex] = [tex]\frac{d}{dh}[/tex]([tex]\frac{1}{3}\pi r^{2} h[/tex]) = [tex]\frac{1}{3}\pi r^{2}[/tex] = 5547π
[tex]\frac{dv}{dt}[/tex] = 11008 π(1.6 in/s) + 5547π (-2.4in/s) = 4300π in³ / s
