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The radius of a spherical balloon is increasing at the rate of 0.6 cm divided by minute. How fast is the volume changing when the radius is 7.5 ​cm? The volume is changing at a rate of nothing cm cubed divided by minute. ​(Type an integer or a decimal. Round to one decimal place as​ needed.)

Respuesta :

Answer:

[tex]\dfrac{dV}{dt} =424.12\ cm^3/min[/tex]

Explanation:

given,                                                          

radius increasing rate = 0.6 cm/minute                    

the volume changing when the radius = 7.5 ​cm                    

volume of the sphere= [tex]\dfrac{4}{3}\pi r^3[/tex]                

                              [tex]\dfrac{dr}{dt} = 0.6 cm/min[/tex]

                       [tex]V =\dfrac{4}{3}\pi r^3[/tex]                                  

                       [tex]\dfrac{dV}{dt} =\dfrac{4}{3}\pi (3r^2)\dfrac{dr}{dt}[/tex]

                       [tex]\dfrac{dV}{dt} =4\pi r^2\dfrac{dr}{dt}[/tex]

                       [tex]\dfrac{dV}{dt} =4\pi 7.5^2\times 0.6[/tex]

                       [tex]\dfrac{dV}{dt} =424.12\ cm^3/min[/tex]

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