Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?
The volume of a cone (V) with radius (r) and height (h) is given by,
\begin{align}V=\frac{1}{3}\pi r^2h\end{align}
It is given that,
\begin{align}h=\frac{1}{6} r\Rightarrow r =6h\end{align}
\begin{align}\therefore V=\frac{1}{3}\pi (6h)^2.h = 12\pi h^3\end{align}
The rate of change of volume with respect to time (t) is given by,
\begin{align} \frac{dV}{dt}=12 \pi \frac{d}{dh}(h^3).\frac{dh}{dt}[By\; Chain\; Rule]\end{align}
\begin{align}=12 \pi (3h^2).\frac{dh}{dt}\end{align}
\begin{align}=36 \pi h^2.\frac{dh}{dt}\end{align}
It is also given that
\begin{align}\frac{dV}{dt}=12\;cm^3/s \end{align}
Therefore, when h = 4 cm, we have:
\begin{align}12=36\pi (4)^2.\frac{dh}{dt}\end{align}
\begin{align}\Rightarrow \frac{dh}{dt}=\frac{12}{36\pi (16)}=\frac{1}{48\pi}\end{align}
Hence, when the height of the sand cone is 4 cm, its height is increasing at the rate of
\begin{align}\frac{1}{48\pi}.\end{align}
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.
Determine order and degree(if defined) of differential equation \begin{align}\frac{d^2y}{dx^2}=\cos3x + sin3x\end{align}
The total revenue in Rupees received from the sale of x units of a product is given by
R (x) = 3x2 + 36x + 5. The marginal revenue, when x = 15 is
(A) 116 (B) 96 (C) 90 (D) 126
In Figure, identify the following vectors.
(i) Coinitial (ii) Equal (iii) Collinear but not equal
The vertices of ΔABC are A (3, 5, −4), B (−1, 1, 2), and C (−5, −5, −2).
Represent graphically a displacement of 40 km, 30° east of north.
The degree of the differential equation
\begin{align}\left(\frac{d^2y}{dx^2}\right)^3\;+ \left(\frac{dy}{dx}\right)^2+\;sin\left(\frac{dy}{dx}\right)\;+ 1=\;0\end{align}
is (A) 3 (B) 2 (C) 1 (D) not defined
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
Classify the following measures as scalars and vectors.
(i) 10 kg (ii) 2 metres north-west (iii) 40°
(iv) 40 watt (v) 10–19 coulomb (vi) 20 m/s2