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.
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Determine order and degree(if defined) of differential equation \begin{align}\left(\frac{ds}{dt}\right)^4\;+\;3s\frac{d^2s}{dt^2}\;=\;0\end{align}
Find gof and fog, if
(i) f(x) = | x | and g(x) = | 5x – 2 |
(ii) f(x) = 8x3 and g(x) = x1/3 .
The total revenue in Rupees received from the sale of x units of a product is given by
R (x) = 13x2 + 26x + 15
Find the marginal revenue when x = 7.
Letbe a function defined as. The inverse of f is map g: Range
(A)
(B)
(C)
(D)
Thanks