Advertisements
Advertisements
प्रश्न
The radius of a circle is increasing at the rate of 0.5 cm/sec. Find the rate of increase of its circumference ?
Advertisements
उत्तर
\[\text { Let r be the radius and C be the circumference of the circle at any time t.Then,}\]
\[C = 2\pi r\]
\[ \Rightarrow \frac{dC}{dt} = 2\pi\frac{dr}{dt}\]
\[ \Rightarrow \frac{dC}{dt} = 2\pi \times 0 . 5 \left[ \because\frac{dr}{dt}=0.5 cm/sec \right]\]
\[ \Rightarrow \frac{dC}{dt} = \pi cm/\sec\]
APPEARS IN
संबंधित प्रश्न
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]
Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm.
A balloon, which always remains spherical, has a variable diameter `3/2 (2x + 1)` Find the rate of change of its volume with respect to x.
The volume of a sphere is increasing at the rate of 8 cm3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.
Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies?
Find the rate of change of the volume of a sphere with respect to its diameter ?
Find the rate of change of the volume of a sphere with respect to its surface area when the radius is 2 cm ?
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 ?
The side of a square sheet is increasing at the rate of 4 cm per minute. At what rate is the area increasing when the side is 8 cm long?
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.
A particle moves along the curve y = x3. Find the points on the curve at which the y-coordinate changes three times more rapidly than the x-coordinate.
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?
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.
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of the area of the rectangle.
The side of a square is increasing at the rate of 0.1 cm/sec. Find the rate of increase of its perimeter ?
Find the surface area of a sphere when its volume is changing at the same rate as its radius ?
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
The altitude of a cone is 20 cm and its semi-vertical angle is 30°. If the semi-vertical angle is increasing at the rate of 2° per second, then the radius of the base is increasing at the rate of
For what values of x is the rate of increase of x3 − 5x2 + 5x + 8 is twice the rate of increase of x ?
The coordinates of the point on the ellipse 16x2 + 9y2 = 400 where the ordinate decreases at the same rate at which the abscissa increases, are
If the rate of change of area of a circle is equal to the rate of change of its diameter, then its radius is equal to
The radius of a circular plate is increasing at the rate of 0.01 cm/sec. The rate of increase of its area when the radius is 12 cm, is
For the curve y = 5x – 2x3, if x increases at the rate of 2 units/sec, then how fast is the slope of curve changing when x = 3?
The rate of change of volume of a sphere with respect to its surface area, when the radius is 2 cm, is ______.
A swimming pool is to be drained for cleaning. If L represents the number of litres of water in the pool t seconds after the pool has been plugged off to drain and L = 200 (10 – t)2. How fast is the water running out at the end of 5 seconds? What is the average rate at which the water flows out during the first 5 seconds?
The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is ______.
A particle is moving along the curve x = at2 + bt + c. If ac = b2, then particle would be moving with uniform ____________.
The radius of a circle is increasing uniformly at the rate of 3 cm per second. Find the rate at which the area of the circle is increasing when the radius is 10 cm.
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?
An edge of a variable cube is increasing at the rate of 10 cm/sec. How fast will the volume of the cube increase if the edge is 5 cm long?
When \(x\) and \(y\) both vary with respect to \(t\), which formula gives the rate of change of \(y\) with respect to \(x\)?
For \[\text{C}(x)=0.005x^3-0.02x^2+30x+5000\], what is the marginal cost when \[x=3\]?
What does a positive derivative mean for a quantity?
What does a negative derivative mean for a quantity?
