Question 8

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Answer

The volume of a sphere (*V)* with radius (*r*) is given by,

\begin{align} V=\frac{4}{3}\pi r^2\end{align}

∴Rate of change of volume (*V)* with respect to time (*t)* is given by,

\begin{align} \frac{dV}{dt} =\frac{dV}{dr}.\frac{dr}{dt}\;\;\;[By\; Chain\; Rule]\end{align}

\begin{align} =\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right).\frac{dr}{dt}\end{align}

\begin{align} =4\pi r^2.\frac{dr}{dt}\end{align}

It is given that

\begin{align} \frac{dV}{dt}=900\; cm^3/s\end{align}

\begin{align} \therefore 900=4\pi r^2.\frac{dr}{dt}\end{align}

\begin{align} \Rightarrow \frac{dr}{dt}=\frac{900}{4\pi r^2}=\frac{225}{\pi r^2}\end{align}

Therefore, when radius = 15 cm,

\begin{align} \frac{dr}{dt}=\frac{225}{\pi (15)^2}=\frac{1}{\pi }\end{align}

Hence, the rate at which the radius of the balloon increases when the radius is 15 cm is

\begin{align} \frac{1}{\pi }\end{align}

- Q:- Given an example of a relation. Which is

(i) Symmetric but neither reflexive nor transitive.

(ii) Transitive but neither reflexive nor symmetric.

(iii) Reflexive and symmetric but not transitive.

(iv) Reflexive and transitive but not symmetric.

(v) Symmetric and transitive but not reflexive. - Q:- Determine whether each of the following relations are reflexive, symmetric and transitive:

(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}

(c) R = {(x, y): x is exactly 7 cm taller than y}

(d) R = {(x, y): x is wife of y}

(e) R = {(x, y): x is father of y} - Q:- Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
- Q:- Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as

R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. - Q:- Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b
^{2}} is neither reflexive nor symmetric nor transitive. - Q:-
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 + x^{2 } - Q:- Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
- Q:-
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. - Q:-
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. - Q:- Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.

- Q:-
The rate of change of the area of a circle with respect to its radius

*r*at*r*= 6 cm is(A) 10π (B) 12π (C) 8π (D) 11π

- Q:-
The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

- Q:-
A balloon, which always remains spherical, has a variable diameter

\begin{align} \frac{3}{2}(2x+1)\end{align}

Find the rate of change of its volume with respect to

*x*. - Q:-
Consider

*f*: R_{+}→ [4, ∞) given by f(x) = x^{2}+ 4. Show that*f*is invertible with the inverse*f*^{–1}of f given by_{}, where R_{+}is the set of all non-negative real numbers. - Q:-
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

- Q:-
The total revenue in Rupees received from the sale of

*x*units of a product is given byR (x) = 3x

^{2}+ 36x + 5. The marginal revenue, when x = 15 is(A) 116 (B) 96 (C) 90 (D) 126

- Q:-
Consider

*f*: R_{+}→ [– 5, ∞) given by*f(x)*= 9x^{2}+ 6x – 5. Show that*f*is invertible

with**.** - Q:- Find the principal value of \begin{align} cos^{-1}\left(\frac{1}{2}\right) + 2sin^{-1}\left(\frac{1}{2}\right)\end{align}
- Q:-
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

- Q:-
If

*f(x)*=_{}, show that*fof*(x) = x, for all x ≠ 2/3. What is the inverse of*f*?

- NCERT Chapter