Advertisements
Advertisements
प्रश्न
A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is ______.
पर्याय
1/10` radian/sec
1/20 radian/sec
20 radian/sec
10 radian/sec
Advertisements
उत्तर
A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is 1/20 radian/sec.
Explanation:
Length of ladder = 5 m
Let AB = y m and BC = x m
∴ In right ΔABC,
AB2 + BC2 = AC2
⇒ x2 + y2 = (5)2
⇒ x2 + y2 = 25
Differentiating both sides w.r.t x, we have
`2x * "dx"/"dt" + 2y * "dy"/"dt"` = 0
⇒ `x "dx"/"dt" + y * "dy"/"dt"` = 0
⇒ `2 * "dx"/"dt" + y xx (-0.1)` = 0 ....[∵ x = 2m]
⇒ `2 * "dx"/"dt" + sqrt(25 - x^2) xx (-0.1)` = 0
⇒ `2 * "dx"/"dt" + sqrt(25 - 4) xx (-0.1)` = 0
⇒ `2 * "dx"/"dt" - sqrt(21)/10` = 0
⇒ `"dx"/"dt" = sqrt(21)/20`
Now cos θ = `"BC"/"AC"` ....(θ is in radian)
⇒ cos θ = `x/5`
Differentiating both sides w.r.t. t, we get
`"d"/"dt" cos theta = 1/5 * "dx"/"dt"`
⇒ `- sin theta ("d"theta)/"dt" = 1/5 * sqrt(21)/20`
⇒ `("d"theta)/"dt" = sqrt(21)/100 xx (- 1/sin theta)`
= `sqrt(21)/100 xx -(1/("AB"/"AC"))`
= `- sqrt(21)/100 xx "AC"/"AB"`
= `- sqrt(21)/100 xx 5/sqrt(21)`
= `- 1/20` radian/sec
[(–) sign shows the decrease of change of angle]
Hence, the required rate = `1/20` radian/sec
APPEARS IN
संबंधित प्रश्न
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?
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.
The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x3 – 0.02x2 + 30x + 5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of change of total cost at any level of output.
The total cost C (x) associated with the production of x units of an item is given by C (x) = 0.007x3 − 0.003x2 + 15x + 4000. Find the marginal cost when 17 units are produced ?
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 10 cm long?
The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate of increase of the perimeter of the square.
The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference?
A man 160 cm tall, walks away from a source of light situated at the top of a pole 6 m high, at the rate of 1.1 m/sec. How fast is the length of his shadow increasing when he is 1 m away from the pole?
A ladder 13 m long leans against a wall. The foot of the ladder is pulled along the ground away from the wall, at the rate of 1.5 m/sec. How fast is the angle θ between the ladder and the ground is changing when the foot of the ladder is 12 m away from the wall.
A particle moves along the curve y = x2 + 2x. At what point(s) on the curve are the x and y coordinates of the particle changing at the same rate?
The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance?
How far is the foot from the wall when it and the top are moving at the same rate?
A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases ?
The volume of metal in a hollow sphere is constant. If the inner radius is increasing at the rate of 1 cm/sec, find the rate of increase of the outer radius when the radii are 4 cm and 8 cm respectively.
A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. Find the rate at which its area is increasing when radius is 3.2 cm.
The volume of a sphere is increasing at 3 cubic centimeter per second. Find the rate of increase of the radius, when the radius is 2 cms ?
The side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?
The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?
Find the surface area of a sphere when its volume is changing at the same rate as its radius ?
If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere ?
If \[V = \frac{4}{3}\pi r^3\] , at what rate in cubic units is V increasing when r = 10 and \[\frac{dr}{dt} = 0 . 01\] ? _________________
The radius of a sphere is changing at the rate of 0.1 cm/sec. The rate of change of its surface area when the radius is 200 cm is
For what values of x is the rate of increase of x3 − 5x2 + 5x + 8 is twice the rate of increase of x ?
The volume of a sphere is increasing at 3 cm3/sec. The rate at which the radius increases when radius is 2 cm, is
The diameter of a circle is increasing at the rate of 1 cm/sec. When its radius is π, the rate of increase of its area is
In a sphere the rate of change of volume is
Water is dripping out at a steady rate of 1 cu cm/sec through a tiny hole at the vertex of the conical vessel, whose axis is vertical. When the slant height of water in the vessel is 4 cm, find the rate of decrease of slant height, where the vertical angle of the conical vessel is `pi/6`
The rate of change of volume of a sphere with respect to its surface area, when the radius is 2 cm, is ______.
A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate
A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.
Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being seperated.
A man, 2m tall, walks at the rate of `1 2/3` m/s towards a street light which is `5 1/3`m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the shadow changing when he is `3 1/3`m from the base of the light?
The rate of change of area of a circle with respect to its radius r at r = 6 cm is ____________.
The rate of change of volume of a sphere is equal to the rate of change of the radius than its radius equal to ____________.
A kite is being pulled down by a string that goes through a ring on the ground 8 meters away from the person pulling it. If the string is pulled in at 1 meter per second, how fast is the kite coming down when it is 15 meters high?
When \(x\) and \(y\) both vary with respect to \(t\), which formula gives the rate of change of \(y\) with respect to \(x\)?
For a circle with area \[A=\pi r^2\], what is \[\frac{dA}{dr}\]?
