Advertisements
Advertisements
प्रश्न
The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side
उत्तर
Let x be the length of the cube
∴ Volume of the cube V = x3 ......(1)
Given that `"dV"/"dt"` = K
Differentiating Equation (1) w.r.t. t, we get
`"dV"/"dt" = 3x^2 * "dx"/"dt"` = K .....(constant)
∴ `"dx"/"dt" = "K"/(3x^2)`
Now surface area of the cube, S = 6x2
Differentiating both sides w.r.t. t, we get
`"ds"/"dt" = 6 * 2 * x * "dx"/"dt"`
= `12x * "K"/(3x^2)`
⇒ `"ds"/"dt" = (4"K")/x`
⇒ `"ds"/"dt" oo 1/x` .....(4K = constant)
Hence, the surface area of the cube varies inversely as the length of the side.
APPEARS IN
संबंधित प्रश्न
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 volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 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 total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.
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.
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.
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?
Find an angle θ which increases twice as fast as its cosine ?
Find an angle θ whose rate of increase twice is twice the rate of decrease of its cosine ?
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.
Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate ?
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.
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 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 ?
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
A man 2 metres tall walks away from a lamp post 5 metres height at the rate of 4.8 km/hr. The rate of increase of the length of his shadow is
In a sphere the rate of change of volume is
In a sphere the rate of change of surface area is
Find the rate of change of the area of a circle with respect to its radius r when r = 4 cm.
Evaluate: `int (x(1+x^2))/(1+x^4)dx`
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 ____________.
Let y = f(x) be a function. If the change in one quantity 'y’ varies with another quantity x, then which of the following denote the rate of change of y with respect to x.
What is the rate of change of the area of a circle with respect to its radius when, r = 3 cm
A spherical balloon is being inflated at the rate of 35 cc/min. The rate of increase in the surface area (in cm2/min.) of the balloon when its diameter is 14 cm, is ______.
A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is ______.
