MOTION IN A STRAIGHT LINE (OR RECTILINEAR MOTION)
EXERCISE - A
2) A particle moves along a line according to the formula, s= t³+ at²+ bt + c, where a,b,c are constants . Find a,b,c if at t= 1s, we have s= 3m, velocity= 7 m/s and acceleration= 12 m/s². 3,-2,1
3) The displacement s in metres of a particle moving in a straight line in t seconds is given by s= 45t + 11t²- t³. Find the time when the particles comes rest. 9sec
4) The distance s metres moved by a particle travelling in a straight line, in t seconds, is given by, s = 3t + t³. Calculate
a) the acceleration after 2 seconds. 12m/s²
b) the time when the velocity is 30 m/s. 3sec
c) the distance moved during the 3rd second. 22m
5) A particle moves in a straight line in such a way that its position at a time t is given by x= 3t + cos3t. Show that its velocity and acceleration both vanish together. Find, how far it moves before coming to rest. π/2
6) The distance x metres covered by a particle moving along a straight line in t seconds is given by formula x= 2t³- 9t²+ 5t +8.
Find when its acceleration is zero. Also, find the velocity at the instant. 1.5 sec, -8.5 m/s
7) A particle moving according to the formula s= 6 + 20t - t², starts from a distance of 6m from a mark and moves along a line farther and farther from the mark. How far from the mark does it go, before it starts moving the opposite direction ? 106m
8) if a particle moves in a straight line according to the formula s= t³- 6t²- 15t; find the time interval during which the velocity of negative and acceleration is positive. t=. 2 to t= 5
9) A particle moves along a line according to the law, x= √(at²+ 2bt + c), where a,b,c are constant. Prove that the acceleration varise inversely as the cube of the distance.
10) A particle moves in a a line so that so that x= √t. Show that the acceleration is negative and proportional to the cube of the velocity.
11) A point moves in a line so that its distance s from a fixed point at any time t is proportional to tⁿ, if v be the velocity and f the acceleration at any time t, show that v²= nfs/(n -1).
12) The law of motion of a particle moving in a line given by s= vt/2, show that the acceleration is constant.
13) A car is running on a straight road. The distance travelled and the time taken are connected by the formula s= (t²- 2t), where s is measured in kilometres and t in hours. When the reading of the odometer is 63, what is the reading of the speedometer? 16
RAW- A
1) A particle is moving in a straight line and the distance s cm covered by it in t seconds is given by the formula, s= 20t - 4t²
Find the distance, velocity and acceleration at the end up 2 seconds. 24cm, 4cm/s, 8 cm/s²
2) A particle, starting from a fixed point O, moves in a straight line. The distance s metres described by it in t seconds is given by, s= 11+ 5t + t². Find
a) the distance from O after 4 seconds. 95m
b) its speed after 4 seconds. 53m/s
c) its acceleration after 4 seconds. 24 m/s²
d) distance travelled by it in the 4-th second. 42m
3) The displacement x of a particle at a time t is given by, x = 2t³- 5t² + 4t+3. Find
a) the time when the acceleration is 8 cm/s². 1.5s
b) velocitya and displacement at the instant. 2.5 cm/s, x= 4.5 cm
4) A particles moves along a straight line according to the law, s= t²- 6t²+ 19t -4. Find
a) its displacement and acceleration when its velocity is 7 m/s. 18m, a= 0
b) its displacement and velocity when its acceleration is 6 m/s². 26m, 10m/s
5) A particle moves a straight line so that after t second its distance from a fixed point O on the line is s meters, where s= t³- 4t²+ 3t. Find
a) when the particle is at O. 0, 1, 3
b) what are its velocity and acceleration at these instants. 3m/s, -2m/s, 6m/s and -8m/s², -2m/s², 10m/s²
6) A particle moves along a line according to the law, s= at²- 2bt + c, where a,b,c different are constants . Prove that the acceleration is constant.
7) If a particle moves along a line so that the distance described is proportional to the square of the time of description, prove that the velocity is proportional to the time and the rate of increase of the velocity will be constant.
8) A car from rest and moves a distance s metres in seconds , where s= a cos t + b sin t. Show that the declaration at time t is the negative of the distance covered by it in t seconds.
9) The distance s in metres described by a particle in t seconds is given by, s= (aeᵗ + be⁻ᵗ). Show that the acceleration of the particle at time t is equal to the distance travelled by it upto time t.
10) A particle moves in a line according to the law s= at²+ bt + c, where a, b, c are constant and s is the distance of the particle from a fixed point O, covered in t seconds. Initially, the particle is 10cm away from O and its initial velocity is 12 cm/s. If the particle moves with a uniform acceleration of 4cm/s², find the distance travelled by it in the 7-th second. 38m
11) The displacement x metres of a particle moving in a straight line at a time t seconds is given by, x = 2t³- 9t²+ 12t +1. find
a) the velocity and acceleration at t= 1 second. 0, -6m/s²
b) the time when the particle stops momentarily. 1s, 2s,
c) the distance between two stops . 6m
12) A particles moves in a line according to the law, s= 6t³+ 20t²+ 9t, where s is in centimetres and t in seconds. Find the initial velocity and acceleration of the particle. 9m/s, 40 cm/s²
13) A particle is moving on a line according to the law, s= tan⁻¹t + at²+ bt + c, where a, b, c are constants. it is given that 1.5 m/s². Find the values a,b,c. 1, 1/2, (8-π)/4
MOTION UNDER GRAVITY
EXERCISE - B
1) A Stone is thrown vertically upwards moving in a line, Its equation of motion is, s= 29.4t - 4.9t². Find the maximum height reached by it. 44.1 units
2) The maximum height is reached in 3 seconds by a stone thrown up vertically and moving under the equation s= ut - 4.9t², where s is in metres and t is in seconds. Find the value of u. 29.4
3) A ball thrown vertically upwards, falls back to the ground after 8 seconds. If the equation of motion is s= ut - 4.9t², where s is in metres and t is in seconds, find the velocity at t= 1. 29.4 m/s
4) Two stones are thrown up simultaneously. The equations of motion are s= 19.6t - 4.9t² and s= 9.8t - 4.9t² for the first and second stone respectively. What is the height of the second stone, when the height of the first stone is maximum ? to the ground
5) Someone standing on a tower of height 19.6 metres , throws a stone vertically upwards. it moves in a vertical line slightly away from the line of the tower and falls on the ground. If its equation of motion is, s= 19.6t - 4.9t², where s is in metres and t is in seconds, how much time does it take for the upward motion and how much for the downward motion ? 2, 4 sec
6) The motion of a ball thrown vertically upwards satisfyies the equation. s= bt½+ ct, Where s and t are measured in metres and seconds respectively. if the maximum height reached by the ball is 4.9 metres and it acceleration is -9.8 m/s², find its height after half a second. 3.675m
7) A stone is thrown upwards on certain planet and its equation of motion is s= 5t - 3t², where s is in metres and t is in seconds. What is the acceleration due to gravity of the planet ? After what time will the stone fall back on the planet again ? 5/3 sec
8) A ball is freely from the top of a tower and during the last second of its motion, it is observed to fall (16/25) of the total height. If the equation of motion is, s= 4.9t², where s in metres and t is in seconds ; find the height of the tower. 30.625 m
RAW- B
1) A stone thrown vertically upwards has its equation of motion, s= 49t - 4.9t¹, where s is in metres and t in seconds. Find the maximum height reached by it. 122.5m
2) A particle is moving in a vertical line as per equation, s= 100t - 4.9t², where s is in metres and t in seconds. Find its velocity at t= 1. At what time is its velocity zero ?
What is the maximum value of s ? 90.2 m/s, 10.2s, 510.2m
3) An arrow shot vertically upwards, movey according to the formula, where s= 49t - 4.9t², wherrs is in metres and t in seconds. Find the time taken by it to reach a height of 117.6 metres .
What is its velocity at the end of 8 second ?
After how much time will it fall on the ground ? 4s,6s, -29.4 m/s, 10s
4) A ball projected vertically upwards has its equation of motion, s= ut - 4.9t², where s is in metres and t in seconds . If the maximum height reached by the ball is 44.1 metres , find the value of u. 29.4
5) A short fired vertically upwards is known to be at a point A at the end of 2 seconds and also again there after 3 more seconds. if the equation of motion is s= ut - 4.9t², where s is in metres and t in seconds; find the height of A above the point of projection. 49m
6) A particle falls down freely from the top of a tower and in the last second of its motion it falls down (9/25) of the total height of the tower. If the equation of motion is s= 4.9t², where s is in metres and t in seconds, find the height of the tower. 122.5 m
DERIVATIVE AS A RATE MEASURE
EXERCISE - C
1) Find the rate of change of the area of a circle of radius r, when the radius varies. 2πr
2) Find the rate of change of the whole surface of a cylinder of radius r and height h, when the radius varies . (4πr+ 2πh)
3) An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10cm long ? 900 cm³/s
4) A sphericals soap bubbles is expanding so that its radius is increasing at the rate of 0.02 cm per second. At what rate is the surface area increasing when its radius is 4cm? (Take π= 3.14). 2.0096 cm²/s
5) The volume of a spherical balloon is increasing at the rate of 20 cm³/s. Find the rate of change of its surface area at the instant when its radius is 8cm. 5 cm²/sec
6) The volume of a cube is increasing at a constant rate. Show that the increase in surface area varies inversely as the length of the edge of the cube.
7) A particle moves along the curve, 6y= (x³+2). Find the points on the curve at which the y-coordinates is changing 8 times as fast as the x-coordinate. (4,11) And (-4, -31/3)
8) 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 m/s. How fast is its height on the wall decreasing, when the foot of the ladder is 4m away from the wall ? (-8/3) m/s
9) A conical vessel whose height is 10m and the radius of whose base is 5 m is being filled with water at the uniform rate of 1.5 cubic metres/minute. Find the rate at which the level of the water in the vessel is rising when the depth is 4m. 3/8π m/min
10) The radius of a cylinder is increasing at the rate of 2m/s and its altitude is decreasing at the rate of 3m/s. Find the rate of change of volume when radius is 3 metres and the altitude 5 metres. 33π cu. m/s
11) A man 160cm tall, walks away from a source of light situated at the top of a pole 6m high, at the rate of 1.1 m/s. How fast is the length of his shadow increasing when he is 1 metre away from the pole ? 0.4 m/s
RAW- C
1) Find the rate of change of the volume of a sphere of radius r with respect to a change in the radius. dV/dr = 4πr²
2) Find the rate of change of the volume of a cylinder of radius r and height h with respect to a change in the radius. dV/dr = 2πrh
3) Find the rate of change of the curved surface of a cone of radius r and height h with respect to a change in the radius . dS/dr = π(h²+ 2r²)/√(h²+ r²)
4) The side of a square is increasing at the rate of 0.2 cm/s. Find the rate of increase of the perimeter of the square. dP/dt = 0.8 cm/s
5) The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference? dC/dt = 4.4 cm/s
6) The radius of a circle is increasing uniformly at the rate of 0.3 cm per second. At what rate is the area increasing when the radius is 10cm ) (π=3.14). dA/dt =18.84cm/s
7) The side of a squares sheet of metal is increasing at 3 cm per minute. At what rate is the area increasing when the side is 10cm long ? dA/dt = 60cm²/min
8) The radius of a circular soap bubble is increasing at the rate of 0.2 cm/s. Find the rate of increase of increase of its surface area, when the radius is 7cm. dS/dt = 35.2 cm²/s
9) The radius of an air bubble is increasing at the rate of 0.5 cm per second. At what rate is the volume of the bubble increasing when the radius is 1 centimetre ? dV/dt = 6.28 cm³/s
10) The volume of a spherical balloon is increasing at the rate of 25 cubic centimetres per second. Find the rate of change of its surface at the instant when its radius is 5cm. dS/dt = 6 cm²/s
11) A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing with the radius is 15cm. (π=22/7). dr/dt = 0.32 cm/s
12) The surface area of a spherical bubble is increasing at 2 cm²/s. When the radius of the bubble is 6cm, at what rate is the volume of the bubble increasing ? dV/dt = 6 cm³/s
13) The bottom of a rectangular swimming tank is 25m by 40m. What is pumped into the tank at the rate of 500 cubic metres per minute. Find the rate at which the level of the water in the tank is rising . dh/dt = 0.5 m/min
14) A stone is dropped into a quite lake and waves move in circle at a speed of 3.5 cm per second. At the instant when the radius of the circular wave is 7.5cm, how fast is the enclosed area increasing. (π=22/7). 165 cm²/s
15) A man 2 meters high, walks at a uniform speed of 5km per hour away from a lamp post, 6 m high. Find the rate at which the length of his shadow increases. 2.5 km/h
16) An inverted cone has a depth of 40cm and a base of radius 5cm. Water is poured into it at a rate of 1.5 cubic centimetres per minute. Find the rate at which the level of water in the cone is rising when depth is 4cm. 1/10π cm/s
17) Sand is pouring from a pipe at the rate of 18cm³/s. The failling sand forms a cone on the ground in such a way that the height of the cone is one-sixth of the radius of the base. How fast is the height of the sand cone increasing when its height is 3 cm ? 1/18π cm/s
18) Water is dripping out from a conical funnel at a uniform rate of 4 cm³/s through a tiny hole at the vertex in the bottom. When the slant height of the water is 3 cm, find the rate of decrease of the slant height of the water, given that the vertical angle of the funnel is 120°. 32/27π cm/s
19) From a Cylindrical drum containing oil and kept vertical, the oil is leacking at the rate of 16ml/s. If the radius of the drum is 7cm and its height is 60cm, find the rate at which the level of oil is changing when oil level is 18cm. 16/49π cm/s
20) A ladder 13m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall , at the rate of 2m/s. How fast is its height on the wall decreasing when the foot of the ladder is 5m away from the wall ? 5/6π cm/s
21) A man is moving away from a tower 40m high at a speed of 2m/s. Find the rate at which the angle of elevation of the top of the tower is changing, when he is at a distance of 30 metres from the foot of the tower. Assume that the eye level of the man is 1.6m from the ground. 0.032 radian/second
22) Find an angle x, which increases twice as fastest as its sine. π/3
23) The radius of a balloon is increasing at the rate of 10 cm/sec. At what rate is the surface area of the balloon increasing when the radius is 15 cm? 1200π cm²/sec
24) An edge a variable cube is increasing at the rate of 5 cm/sec. How fast is the volume of the cube increasing when the edge is 10cm long? 1500 cm³/sec
ERRORS AND APPROXIMATION
1) Find the approximate value of the cube root of 127. 377/75
2) Find the approximate value of (0.007)¹⁾³. 23/120
3) Find the approximate value of square root of 401. 20.025
4) Find the approximate value of √0.037. 0.1925
5) Find the value of log₁₀10.1 , where log₁₀e= 0.4343. 1.004343
6) Find the approximate value of tan46, if it is being given that 1°=0.01745 radians. 1.03490
7) if the radius of a circle increases from 5cm to 5.1 cm, find the increase in area. π cm²
8) if y= (x⁴- 12) and if x changes from 2 to 1.99. What is the approximate change in y ? -0.32
9) The time T of oscillation of a simple pendulum of length l is given by T= 2π √(l/g). Find the percentage error in T corresponding to an error of 2% in the value of l. 1%
10) if the error committed in measuring the radius of a circle be 0.01%, find the corresponding error in calculating the area. 0.02%
11) If in a Triangle ABC , the side c and the angle C remain constant, while the remaining elements are changed slightly, show that da/cosA + db/cosB =0.
12) The area S of a triangle is calculated is measuring b, c and A. if there be an error δA in the measurement of A, show that the relative error in area is given by δS/S = cosA. δA.
RAW-
1) (26)¹⁾³. 80/27
2) (123)¹⁾³. 373/75
3) (7.5)¹⁾³. 1.96
4) (0.009)¹⁾³. 0.208
5) √25.2. 5.02
6) √0.163. 0.404
7) (255)¹⁾⁴. 3.9961
8) (15)¹⁾⁴. 63/32
9) Find the approximate value of 1/(2.002)². 0.2495
10) Calculate the approximate value of log₁₀(4.04), it being given that log₁₀4= 0.6021 and log₁₀e= 0.4343. 0.606443
11) Find the value of logₑ10.02, it being given that logₑ10 = 2..3026. 2.3046
12) if y= sin x and x changes from π/2 to 22/14, what is the approximate change in y ? No change
13) A circular metal plate under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm. 4π cm²
14) Find the approximate the value of cos 61°, it being given that sin 60°= 0.86603 and 1°= 0.01745 radians . 0.4849
15) if the length of a simple pendulum is decreased by 2%, find the percentage decrease in its period T, where T= 2π √(l/g). 1%
16) The pressure p and the volume V of a gas are connected by the relation, pt¹⁾⁴= k, where k is a constant. Find the percentage increase in the pressure corresponding to a diminution of 0.5% in the volume. 0.125%
17) The radius of a sphere shrinks from 10cm to 9.8cm. Find the approximate the decrease in
a) volume . 80π cu.cm
b) surface area. 16π cm²
18) if there is an error of 0.1% in the measurement of the radius of a sphere, find the approximately the percentage error in the calculation of the volume of the sphere. 0.3%
19) Show that the relative error in the volume of sphere, due to error in measuring the diameter, is 3 times the relative error in the diameter.
MAXIMUM AND MINIMUM
EXERCISE - A
1) Find all the points of local maxima and minimum and the corresponding maximum and minimum values of the function f(x)= -3x⁴/4 - 8x³ - 45x²/2 + 105. 0, (105), -5, (295/4), -3,(231/4)
2) Find the local maximum and the local minimum the function
a) f(x)= sin2x, where 0< x < π. π/4,(1), 3π/4,(-1)
b) f(x)= (sin2x - x), where -π/2≤ x ≤π/2. Min:-π/6,(-(√3/2)+π/6) max: π/6,(√3/2-π/6)
3) Find the local maximum and the local minimum of the function
a) f(x)= (sinx - cosx), where 0< x<π/4. Max:3π/4,(√2), min:7π/4,(-√2
b) f(x)= (2cosx + x), where 0< x <π. Max:π/6,(√3+π/6), min: 5π/6,(5π/6 -√3)
4) Find the point of local Maxima or local minimum the function
f(x)= (sin⁴x + cos⁴x) in 0< x <π/2. Min: π/4,(1/2)
5) Find the local Maxima and local minimum and the corresponding local maximum and local minimum value of the following functions:
a) f(x)= x √(1- x), where x > 0. Max: 2/3,(2/3√3
b) f(x)= x/{(x -1)(x -4)}, where 1< x < 4. Max: 2,(-1)
6) Prove that the maximum value of (1/x)ˣ is e¹⁾ᵉ.
7) Find the point of the parabola y²= 2x which is closest to the point (1,4). (2,2)
8) Show that the none of the following function has a maximum or minimum
a) eˣ
b) logx
c) x³+ x²+ x +1.
9) Show that sinᵖθ cosʳθ attain a maximum, when θ = tan⁻¹√(p/r).
10) Find the maximum and minimum values of
(3x⁴- 8x³+ 12x² - 48x + 25) on [0,3]. L Min:2, min.v in [0,3] is 39 at x=2, max.v: in [0,3] is 25 at x=0
11) Find the maximum and minimum value of (x + sin2x) in [0,2π].max.v:2π at 0, min.v: at2π/3 (4π-3√3)/6
12) Show that sinx (1- cosx), x ∈ [0,π] is maximum at x= π/3
RAW- A
1) Find the maximum or minimum values of the following:
a) (5x -1)²+ 4. Min.v: 4
b) -(x -3)²+9. Max.v: 9
c) -|x +4|+ 6. Max: 6
d) sin2x +5. Max: 4, min: 6
e) |sin4x + 3|. Mx: 4, min: 2
2) Find the points of local maximum or local minimum and the corresponding local maximum and the minimum value of each of the following functions.
a) f(x)= (x -3)⁴. Mx is 0 at 3
b) f(x)= = x². Min is 0 at x=0
c) f(x)= 2x³ - 21x²+ 36x -20. Max is -3 at x=1; min is -128 at x=6
d) f(x)= x³- 6x²+ 9x +15. Max is 19 at x=1, min is 15 at x=3
e) f(x)= x⁴- 62x²+ 120x +9. Max is 68 at x=1, min are -1647 at x=-6 and -316 at x= 5
f) f(x)= - x³+ 12x²- 5. Max is 251 at x=8 , min is -5 at x=0
g) f(x)= (x -1)(x +2)². Max is 0 at x=-2, min is -4 at x=0
h) f(x)= -(x -1)³(x +1)². Max is 0 at x=1 and x= -1 , min is -3456/3125 at x= -1/5
i) f(x)= x/2 + 2/x, x>0. Min is 2 at x=2
3) Find the maximum and minimum values of (2x³- 24x +107) in the interval [-3,3]. Max is 139 at x=-2 , min is 189 at x=3
4) Find the maximum and minimum values of 3x⁴- 8x³+ 12x²- 48x +1 on the integral [1,4]. Max is 237 at x=4, min is -63 at x= 2
5) Find the maximum and the minimum values of
f(x)= {sinx + (1/2) cosx} in 0≤ x ≤π/2. Max is 3/4 at x=π/6, min is 1/2 at x= π/2
6) Show that the maximum value of x¹⁾ˣ is e¹⁾ˣ.
7) Show that (x + 1/x) has a maximum and minimum, but the maximum value is less than the minimum value.
8) Find the maximum profit that a company can make, if the profit function is given by p(x)= 41+ 24x - 18x². 49
9) An enemy jet is flying along the curve y= (x²+2). A soldier is placed at the point (3,2). Find the nearest point between the soldier and the jet. (1,3)
10) Find the maximum and minimum values of f(x)= (-x + 2 sinx) in [0,2π]. Max is (-π/3+ √3) at x=π/3, min is -(5π/3+ √3) at x= 5π/3
EXERCISE - B
1) Amongst all pairs of positive numbers with sum 24, find these whose product is maximum. 12,12
2) Amongst all pairs of positive numbers with product 256, find those whose sum is the least. 16,16
3) Find two positive numbers X and y such that, (x + y)= 60 and xy³ is maximum. 45,15
4) Find two numbers whose sum is 16 and the sum of whose cubes is minimum. 8,8
5) Show that all the rectangles with a given perimeter, the square has the largest area.
6) Show that, of all the rectangles of a given area, the square has the smallest perimeter.
7) Prove that the area of a right angled triangle of a given hypotenuse is maximum when the triangle is isosceles.
8) If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that area of the triangle is maximum when the angle between them is (π/3).
9) Two sides of a triangle are given. Find the angle between them such that the area shall be maximum.
10) Show that, of all the rectangle inscribed in a given fixed circle, the square has the maximum area.
11) Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.
12) The combined resistance R of a two resistors R₁ and R₂ where R₁, R₂ > 0 is given by 1/R = 1/R₁ + 1/R₂. If R₁+ R₂ = C(constant), show that the maximum resistance R is obtained by choosing R₁ = R₂.
13) A beam of length l is supported at one end. if W is the uniform load per unit length, the bending moment M at a distance x from the end is given by. M=(lx/2 - Wx²/2). Find the point on the beam at which the bending moment has the maximum value. 1/2W
14) A wire of length 25 m is to be cut into two pieces. One of the wires is to be made into square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum ? 100/(π+4), 25π/(π+4)
15) Show that a cylinder of a given volume which is open at the top has minimum total surface area, provided its height is equal to the radius of its base.
16) Show that the height of a cylinder, which is open at the top, having a given surface and the greatest volume, is equal to the radius of its base.
17) Show that the semi vertical angle of a cone of maximum volume and of a given slant height is tan⁻¹√2.
18) Show that the semi vertical angle of a right circular cone of given surface area and maximum volume is sin⁻¹(1/3).
19) Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.
20) Show that the height of a closed cylinder of given surface and maximum volume, is equal to the diameter of its base.
21) A closed right circular cylinder has volume 2156 cubic units. What should be the radius of its base, so that its total surface area may be maximum ?
22) Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3. Find the volume of the largest cylinder inscribed in a sphere of radius R.
23) Show that the cone of the greatest volume which can be inscribed in a given sphere is such that three times its altitude is twice the diameter of the sphere. Find the volume of the largest cone inscribed in a sphere of radius R.
24) prove that the parameter of a right angled triangle of given hypotenuse is maximum when the triangle is isosceles.
25) An open box is to be made out of a piece of cardboard measuring (24cm x 24 cm) by cutting off equal squares from the corners in the turning of the side. Find the height of the box having maximum volume. 4cm
