The radius of an air bubble is increasing at the rate of 1/2 cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
The air bubble is in the shape of a sphere.
Now, the volume of an air bubble (V) with radius (r) is given by,
\begin{align} V = \frac{4}{3}\pi r^3 \end{align}
The rate of change of volume (V) with respect to time (t) is given by,
\begin{align} \frac{dV}{dt} = \frac{4}{3}\pi \frac{d}{dr}(r^3).\frac{dr}{dt} \;\;\;[By\; Chain\; Rule] \end{align}
\begin{align} = \frac{4}{3}\pi (3r^2).\frac{dr}{dt} \end{align}
\begin{align} = \frac{4}{3}\pi r^2.\frac{dr}{dt} \end{align}
It is given that
\begin{align} \frac{dr}{dt}=\frac{1}{2} cm/s .\end{align}
Therefore, when r = 1 cm,
\begin{align} \frac{dV}{dt}=4\pi(1)^2.(\frac{1}{2})=2\pi\; cm^3/s \end{align}
Hence, the rate at which the volume of the bubble increases is 2π cm3/s.
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R = {(x, y): y is divisible by x}
(iv) Relation R in the set Z of all integers defined as
R = {(x, y): x − y is as integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
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(i) Relation R in the set A = {1, 2, 3,13, 14} defined as
R = {(x, y): 3x − y = 0}
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}
(iv) Relation R in the set Z of all integers defined as
R = {(x, y): x − y is as integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
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(d) R = {(x, y): x is wife of y}
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