The radius of a spherical ball is changing with time. The rate of change of its volume is given be
a)(4πr^2)dr/dt
b)4/3(πr^2)
c)8/3(πr^2)
d)(8πr/3)dr/dt
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Answered by
2
The relationship between a where's volume and it's radius is
V=43πr3
As long as this geometric relationship doesn't change as the sphere grows, then we can derive this relationship implicitly, and find a new relationship between the rates of change.
Implicit differentiation is where we derive every variable in the formula, and in this case, we derive the formula with respect to time.
So we take the derivative of our sphere:
V=43πr3
dVdt=43π(3r2)drdt
dVdt=4πr2drdt
We were actually given drdt. It's 4cms.
We are interested in the moment when the diameter is 80 cm, which is when the radius will be 40 cm.
The rate of increase of the volume is dVdt, which is what we are looking for, so:
dVdt=4πr2drdt
V=43πr3
As long as this geometric relationship doesn't change as the sphere grows, then we can derive this relationship implicitly, and find a new relationship between the rates of change.
Implicit differentiation is where we derive every variable in the formula, and in this case, we derive the formula with respect to time.
So we take the derivative of our sphere:
V=43πr3
dVdt=43π(3r2)drdt
dVdt=4πr2drdt
We were actually given drdt. It's 4cms.
We are interested in the moment when the diameter is 80 cm, which is when the radius will be 40 cm.
The rate of increase of the volume is dVdt, which is what we are looking for, so:
dVdt=4πr2drdt
Answered by
3
a) option is right answer
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