Advertisements
Advertisements
प्रश्न
A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is dragged away from the wall along the ground at the rate of 2 cm/sec. How fast is the height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
Advertisements
उत्तर
Let:
x = distance from the wall to the foot of the ladder (in meters)
y = height of the ladder on the wall (in meters)
The length of the ladder is constant: L = 13 m
Apply the Pythagorean theorem
x2 + y2 = L2
Substituting L = 13:
x2 + y2 = 132 = 169
Differentiate with respect to time
Differentiating both sides of the equation with respect to time t:
`d/(dt) (x^2) + d/(dt)(y^2) = d/(dt) (169)`
This gives us:
`2x(dx)/(dt) + 2y(dy)/(dt) = 0`
Solve for `(dy)/(dt)`
Rearranging the equation:
`2x(dx)/(dt) + 2y(dy)/(dt) = 0`
⟹ `x(dx)/(dt) + y(dy)/(dt) = 0`
Thus: `(dy)/(dt) = -x/y (dx)/(dt)`
Substitute known values
We know: `(dx)/(dt)` = 2 m/sec (the rate at which the foot of the ladder is moving away from the wall)
When x = 5 m, we need to find y.
Using the Pythagorean theorem to find y when x = 5:
52 + y2 = 169
⟹ 25 + y2 = 169
⟹ y2 = 144
⟹ y = 12 m
Now substitute x = 5, y = 12, and `(dx)/(dt) = 2` into the equation for `(dy)/(dt)`
`(dy)/(dt) = -5/12 * 2 = -10/12 = -5/6` m/sec
The negative sign indicates that the height is decreasing.
Therefore, the height on the wall is decreasing at a rate of `5/6` m/sec
APPEARS IN
संबंधित प्रश्न
If y = f (u) is a differential function of u and u = g(x) is a differential function of x, then prove that y = f [g(x)] is a differential function of x and `dy/dx=dy/(du) xx (du)/dx`
A point source of light is hung 30 feet directly above a straight horizontal path on which a man of 6 feet in height is walking. How fast will the man’s shadow lengthen and how fast will the tip of shadow move when he is walking away from the light at the rate of 100 ft/min.
The rate of growth of bacteria is proportional to the number present. If, initially, there were
1000 bacteria and the number doubles in one hour, find the number of bacteria after 2½
hours.
[Take `sqrt2` = 1.414]
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.
A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
The radius of an air bubble is increasing at the rate `1/2` cm/s. At what rate is the volume of the bubble increasing when the radius is 1 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.
Find the rate of change of the area of a circular disc with respect to its circumference when the radius is 3 cm ?
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 when the radius is 15 cm.
The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm?
A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area increasing?
A man 180 cm tall walks at a rate of 2 m/sec. away, from a source of light that is 9 m above the ground. How fast is the length of his shadow increasing when he is 3 m away from the base of light?
A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height h, when h = 9 cm.
Water is running into an inverted cone at the rate of π cubic metres per minute. The height of the cone is 10 metres, and the radius of its base is 5 m. How fast the water level is rising when the water stands 7.5 m below the base.
The surface area of a spherical bubble is increasing at the rate of 2 cm2/s. When the radius of the bubble is 6 cm, at what rate is the volume of the bubble increasing?
Sand is being poured onto a conical pile at the constant rate of 50 cm3/ minute such that the height of the cone is always one half of the radius of its base. How fast is the height of the pile increasing when the sand is 5 cm deep ?
A kite is 120 m high and 130 m of string is out. If the kite is moving away horizontally at the rate of 52 m/sec, find the rate at which the string is being paid out.
If a particle moves in a straight line such that the distance travelled in time t is given by s = t3 − 6t2+ 9t + 8. Find the initial velocity of the particle ?
The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?
Side of an equilateral triangle expands at the rate of 2 cm/sec. The rate of increase of its area when each side is 10 cm is
A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π m3/minute. The rate at which the surface of the oil is rising, is
The distance moved by the particle in time t is given by x = t3 − 12t2 + 6t + 8. At the instant when its acceleration is zero, the velocity is
The radius of a sphere is increasing at the rate of 0.2 cm/sec. The rate at which the volume of the sphere increase when radius is 15 cm, is
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 equation of motion of a particle is s = 2t2 + sin 2t, where s is in metres and t is in seconds. The velocity of the particle when its acceleration is 2 m/sec2, is
Find the rate of change of the area of a circle with respect to its radius r when r = 4 cm.
The radius of a cylinder is increasing at the rate of 3 m/s and its height is decreasing at the rate of 4 m/s. The rate of change of volume when the radius is 4 m and height is 6 m, is ____________.
The rate of change of area of a circle with respect to its radius r at r = 6 cm is ____________.
A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?
