A balloon, which always remains spherical, has a variable diameter Find the rate of change of its volume with respect to x .
The diameter of spherical balloon is variable.
Let the radius of spherical balloon be r , diameter be d and the volume of sphere be V .
The variable diameter of spherical balloon is governed by the equation,
d = 3 2 ( 2 x + 1 )
The radius of spherical balloon is governed by the equation,
r = d 2
Substitute the value of diameter in the above expression.
r = 3 2 ( 2 x + 1 ) 2 = 3 4 ( 2 x + 1 )
The volume of the air bubble is given by,
V = 4 3 π r 3 (1)
Substitute the value of r in equation (1).
V = 4 3 π [ 3 4 ( 2 x + 1 ) ] 3 = 4 3 π × ( 3 4 ) 3 × ( 2 x + 1 ) 3 = 9 16 π ( 2 x + 1 ) 3 (2)
Differentiate equation (2) to obtain the rate of change of V with respect to x .
d V d x = 9 16 π d d x ( 2 x + 1 ) 3 = 9 16 π × 3 × ( 2 x + 1 ) 2 × 2 = 9 8 × 3 × π × ( 2 x + 1 ) 2 = 27 8 π ( 2 x + 1 ) 2
Thus, the volume of bubble is increasing at the rate of 27 8 π ( 2 x + 1 ) 2 .
The diameter of spherical balloon is variable.
Let the radius of spherical balloon be
The variable diameter of spherical balloon is governed by the equation,
The radius of spherical balloon is governed by the equation,
Substitute the value of diameter in the above expression.
The volume of the air bubble is given by,
Substitute the value of
Differentiate equation (2) to obtain the rate of change of
Thus, the volume of bubble is increasing at the rate of
Find the rate of change of its volume with respect to