Question 1

Q1: Find the rate of change of the area of a circle with respect to its radius $r$ when (a) $r=3 \mathrm{~cm}$ (b) $r=4 \mathrm{~cm}$

Accepted Answer

Given: The area of a circle is a function of its radius $r$. To Find: The rate of change of the area of the circle with respect to its radius $r$ for given values of $r$. Formula: The area A of a circle with radius $r$ is given by: $$A = \pi r^2$$ Solution: We need to find the rate of change of area...

Question 2

Q10: A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of $2 \mathrm{~cm} /...

Accepted Answer

Given: Length of the ladder = $5 \mathrm{~m}$. Rate at which the bottom of the ladder is pulled away from the wall, $\frac{dx}{dt} = 2 \mathrm{~cm}/\mathrm{s} = 0.02 \mathrm{~m}/\mathrm{s}$. Distance of the foot of the ladder from the wall, $x = 4 \mathrm{~m}$. To Find: The rate at which the heig...

Question 3

Q2: The volume of a cube is increasing at the rate of $8 \mathrm{~cm}^{3} / \mathrm{s}$. How fast is the surface area increasing when the length of an edg...

Accepted Answer

Given: Rate of increase of volume of a cube, $\frac{dV}{dt} = 8 \mathrm{~cm}^3/\mathrm{s}$. Length of an edge, $x = 12 \mathrm{~cm}$. To Find: The rate at which the surface area is increasing, $\frac{dS}{dt}$. Let: $x$ be the length of an edge of the cube. $V$ be the volume of the cube. $S$ be th...

Question 4

Q3: The radius of a circle is increasing uniformly at the rate of $3 \mathrm{~cm} / \mathrm{s}$. Find the rate at which the area of the circle is increasi...

Accepted Answer

Given: Rate of increase of radius, $\frac{dr}{dt} = 3 \mathrm{~cm}/\mathrm{s}$. Radius, $r = 10 \mathrm{~cm}$. To Find: The rate at which the area of the circle is increasing, $\frac{dA}{dt}$. Let: $r$ be the radius of the circle. $A$ be the area of the circle. Formula: Area of a circle: $A = \p...

Question 5

Q4: An edge of a variable cube is increasing at the rate of $3 \mathrm{~cm} / \mathrm{s}$. How fast is the volume of the cube increasing when the edge is...

Accepted Answer

Given: Rate of increase of the edge of a cube, $\frac{dx}{dt} = 3 \mathrm{~cm}/\mathrm{s}$. Length of the edge, $x = 10 \mathrm{~cm}$. To Find: The rate at which the volume of the cube is increasing, $\frac{dV}{dt}$. Let: $x$ be the length of an edge of the cube. $V$ be the volume of the cube. Fo...

Question 6

Q5: A stone is dropped into a quiet lake and waves move in circles at the speed of $5 \mathrm{~cm} / \mathrm{s}$. At the instant when the radius of the ci...

Accepted Answer

Given: Rate of increase of the radius of the circular wave, $\frac{dr}{dt} = 5 \mathrm{~cm}/\mathrm{s}$. Radius of the circular wave, $r = 8 \mathrm{~cm}$. To Find: The rate at which the enclosed area is increasing, $\frac{dA}{dt}$. Let: $r$ be the radius of the circular wave. $A$ be the enclosed...

Question 7

Q6: The radius of a circle is increasing at the rate of $0.7 \mathrm{~cm} / \mathrm{s}$. What is the rate of increase of its circumference?

Accepted Answer

Given: Rate of increase of radius, $\frac{dr}{dt} = 0.7 \mathrm{~cm}/\mathrm{s}$. To Find: The rate of increase of the circumference, $\frac{dC}{dt}$. Let: $r$ be the radius of the circle. $C$ be the circumference of the circle. Formula: Circumference of a circle: $C = 2\pi r$ Solution: Differen...

Question 8

Q7: The length $x$ of a rectangle is decreasing at the rate of $5 \mathrm{~cm} /$ minute and the width $y$ is increasing at the rate of $4 \mathrm{~cm} /$...

Accepted Answer

Given: Rate of change of length, $\frac{dx}{dt} = -5 \mathrm{~cm}/\mathrm{min}$ (decreasing). Rate of change of width, $\frac{dy}{dt} = 4 \mathrm{~cm}/\mathrm{min}$ (increasing). Instantaneous length, $x = 8 \mathrm{~cm}$. Instantaneous width, $y = 6 \mathrm{~cm}$. To Find: (a) The rate of chang...

Question 9

Q8: 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 whic...

Accepted Answer

Given: Rate of increase of volume, $\frac{dV}{dt} = 900 \mathrm{~cm}^3/\mathrm{s}$. Radius of the balloon, $r = 15 \mathrm{~cm}$. To Find: The rate at which the radius is increasing, $\frac{dr}{dt}$. Let: $r$ be the radius of the spherical balloon. $V$ be the volume of the balloon. Formula: Volum...

Question 10

Q9: A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10...

Accepted Answer

Given: Radius of the balloon, $r = 10 \mathrm{~cm}$. To Find: The rate at which the volume is increasing with respect to the radius, $\frac{dV}{dr}$. Let: $r$ be the radius of the spherical balloon. $V$ be the volume of the balloon. Formula: Volume of a sphere: $V = \frac{4}{3}\pi r^3$ Solution:...

NCERT Solutions