CBSE Mathematics for Class XII

Dinesh Khattar; Anita Khattar

Chapter 6

Applications of Derivatives - all with Video Answers

Educators

Section 1

Rates Of Change Of Quantities

Problem 1

Find the rate of change of the area of a circle with respect to its radius $r$ when $r=5 \mathrm{~cm}$.

Tanishq Gupta

Tanishq Gupta

Numerade Educator

Problem 2

How fast is the volume of a ball changing with respect to its radius when the radius is $3 \mathrm{~m}$ ?

Christy Galilei

Christy Galilei

Numerade Educator

Problem 3

The radius of a circle is increasing uniformly at the rate of $3 \mathrm{~cm}$ per second. Find the rate at which the area of the circle is increasing when the radius is $10 \mathrm{~cm}$.

Tanishq Gupta

Numerade Educator

Problem 4

The volume of a cube is increasing at a rate of 9 cubic cm per second. How fast is the surface area increasing when the length of an edge is $10 \mathrm{~cm} ?$

Tanishq Gupta

Numerade Educator

Problem 5

Water is being pumped into a conical reservoir of height $10 \mathrm{~m}$ and radius $5 \mathrm{~m}$ at a constant rate of 2 cu m/minute. How fast does the water level rise from a depth of $2 \mathrm{~m}$ ?

Will Erickson

Numerade Educator

Problem 6

A balloon which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the latter is $10 \mathrm{~cm}$.

Tanishq Gupta

Numerade Educator

Problem 7

The radius of a circle is increasing at $0.7 \mathrm{~cm} / \mathrm{s}$. What is the rate of increase of its circumference when $r=4.9 \mathrm{~cm}$ ?

Tanishq Gupta

Numerade Educator

Problem 8

The radius of an air bubble is increasing at the rate of $\frac{1}{2} \mathrm{~cm} / \mathrm{s} .$ At what rate is the volume of the bubble increasing when the radius is $1 \mathrm{~cm}$ ?

Dharmendra Jain

Dharmendra Jain

Numerade Educator

Problem 9

A balloon which always remains spherical has a variable diameter $\frac{3}{2}(2 x+3)$. Find the rate of change of its volume with respect to $x$.

Tanishq Gupta

Numerade Educator

Problem 10

Find the rate of change of the area of a circle with respect to its radius $r$. How fast is the area changing with respect to the radius when the radius is $3 \mathrm{~cm}$ ?

Tanishq Gupta

Numerade Educator

Problem 11

Find the rate of change of the volume of a ball with respect to its radius $r$. How fast is the volume changing with respect to the radius when the radius is $2 \mathrm{~m}$ ?

Carson Merrill

Numerade Educator

Problem 12

The radius of a circle is increasing uniformly at the rate of $4 \mathrm{~cm}$ per second. Find the rate at which the area of the circle is increasing when the radius is $8 \mathrm{~cm}$.

Caleb Fink

Numerade Educator

Problem 13

The volume of a cube is increasing at a rate of 7 cubic $\mathrm{cm}$ per second. How fast is the surface area increasing when the length of an edge is $12 \mathrm{~cm} ?$

Tanishq Gupta

Numerade Educator

Problem 14

The length $x$ of a rectangle is decreasing at the rate of $2 \mathrm{~cm} / \mathrm{s}$ and the width $y$ is increasing at the rate of $2 \mathrm{~cm} / \mathrm{s}$. When $x=12 \mathrm{~cm}$ and $y=5 \mathrm{~cm}$, find the rate of change of (i) the perimeter and (ii) the area of the rectangle.

Israel Hernandez

Israel Hernandez

Numerade Educator

Problem 15

A man is moving away from a tower $85 \mathrm{~m}$ high at a speed of $4 \mathrm{~m} / \mathrm{s}$. Find the rate at which his angle of elevation of the top of the tower is changing when he is at distance of $60 \mathrm{~m}$ from the foot of the tower.

Angela Guo

Numerade Educator

Problem 16

A small stone thrown into a still pond produces a circular disturbance on the surface of water whose radius is increasing at the rate of $2 \mathrm{~cm}$ per second. Find the rate at which the disturbed area is increasing when its radius is $7 \mathrm{~cm}$.

Daphne Pusey

Numerade Educator

Problem 17

At the point $(2,5)$ on the curve $y=x^{3}-2 x+9$, show that the gradient is increasing 12 times as fast as $x$.

Lucas Finney

Numerade Educator

Problem 18

If $y=7 x-x^{3}$ and $x$ increases at the rate of 4 units per second. How fast is the slope of the curve changing when $x=2 ?$

Sunanda Adibhatla

Sunanda Adibhatla

Numerade Educator

Problem 19

Water is running out of a cistern in the form of an inverted right circular cone of angle $45^{\circ}$ with its axis vertical. Find the rate at which the water is flowing out at the instant when depth of water is $2 \mathrm{ft}$, given that at that instant the level of water is diminishing at the rate 3 inches $/ \mathrm{min}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 20

Water is dripping out at the steady rate of $1 \mathrm{cc} / \mathrm{s}$ through a tiny hole at the vertex of a conical vessel, whose axis is vertical. When the slant height of the water in the filter is $4 \mathrm{~cm}$, find the rate of increase of
(i) the slant height of water.
(ii) the area of water surface, given that the vertical angle of vessel is $60^{\circ}$.

Nick Johnson

Numerade Educator

Problem 21

(i) The base radius of a cylindrical vessel full of oil is $30 \mathrm{~cm}$. Oil is drawn at the rate of $27,000 \mathrm{~cm}^{3} / \mathrm{min}$. Find the rate at which the level of oil is falling in the vessel.
(ii) The radius of a cylinder is increasing at the rate of $2 \mathrm{~cm} / \mathrm{s}$ and its altitude is decreasing at the rate of $3 \mathrm{~cm} / \mathrm{s}$. Find the rate of change of volume, when the radius is $3 \mathrm{~cm}$ and altitude is $5 \mathrm{~cm}$.

Carlos Pinilla

Numerade Educator

Problem 22

A ladder $20 \mathrm{ft}$ long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. Show that when the lower end of the ladder is $16 \mathrm{ft}$ away from the wall, upper end is moving $4 / 3$ times as fast as the lower end.

Gio Maya

Numerade Educator

Problem 23

(i) A kite is flying at a height of $40 \mathrm{~m}$. The boy flying it is carrying it horizontally at the rate of $3 \mathrm{~m} / \mathrm{s}$. At what rate is the string being paid out when the length of the string is $50 \mathrm{~m}$. (Assume that the height of the kite remains the same and the string is straight.)
(ii) A man on a wharf $20 \mathrm{~m}$ above the water level, pulls a rope to which a boat is attached at the rate of $4 \mathrm{~m} / \mathrm{s}$. At what rate is the boat approaching the shore, when there is still $25 \mathrm{~m}$ of rope out?

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 24

(i) A man $2 \mathrm{~m}$ high walks at a uniform speed of $5 \mathrm{kmph}$, away from a lamp post $6 \mathrm{~m}$ high. Find the rate at which the length of his shadow increase.
(ii) A man $160 \mathrm{~cm}$ tall, walks away from a source of light situated at the top of a pole $6 \mathrm{~m}$ high, at the rate of $1.1 \mathrm{~m} / \mathrm{s}$. How fast the length of his shadow increases when he is $1 \mathrm{~m}$ away from the post.

Audrey Fong

Numerade Educator

Problem 25

(i) A particle moves along the curve $y=\frac{2}{3} x^{3}+1$. Find the points on the curve at which the $y$ coordinate is changing twice as fast as the $x$ coordinate.
(ii) A particle moves along the curve $y=\frac{4}{3} x^{3}+5$. Find the points on the curve at which the $y$ coordinate changes as fast as the $x$ coordinate.
(iii) For the function $y=x^{3}+21$, find the value of $x$ when $y$ increases 75 times as fast as $x$.
(iv) At what points of the ellipse $16 x^{2}+9 y^{2}=400$, does the ordinate decrease at the same rate which the abscissa increases?

Madi Sousa

Numerade Educator

Problem 26

(i) A hemisphere is constructed on a circular base. If radius of base is increasing at the rate of $0.5 \mathrm{~cm} / \mathrm{s}$, find the rate at which volume of hemisphere is increasing when radius is $10 \mathrm{~cm} ?$
(ii) The bottom of a rectangular swimming tank is $25 \mathrm{~m} \times 40 \mathrm{~m}$. Water is pumped into the tank at the rate of $500 \mathrm{~m}^{3} / \mathrm{h}$. Find the rate at which level of water in the tank is rising.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 27

An inverted cone has a depth of $40 \mathrm{~cm}$ and a base of radius $5 \mathrm{~cm}$. Water is poured into it at a rate of $1.5 \mathrm{~cm}^{3} / \mathrm{min}$. Find the rate at which the level of water in the cone is rising when depth is $4 \mathrm{~cm}$.

Angela Guo

Numerade Educator

Problem 28

From a cylindrical drum containing oil and kept vertical, the oil is leaking at the rate of $16 \mathrm{~cm}^{3} / \mathrm{sec}$. If the radius of the drum is $7 \mathrm{~cm}$ and its height is $60 \mathrm{~cm}$, find the rate at which the level of oil is changing when oil level is $18 \mathrm{~cm}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator