Advertisements
Advertisements
Question
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?
Advertisements
Solution
\[\text { Given }: A = x^2 \text { and } \frac{dx}{dt}= 4 cm/min \]
\[\text { Let x be the side of the square and A be its area at any time t.Then },\]
\[A = x^2 \]
\[ \Rightarrow \frac{dA}{dt} = 2x\frac{dx}{dt}\]
\[ \Rightarrow \frac{dA}{dt} = 2 \times 8 \times 4 \left[ \because x = 8 cm\text { and } \frac{dx}{dt} = 4 cm/\min \right]\]
\[ \Rightarrow \frac{dA}{dt} = 64 \ {cm}^2 /\min\]
APPEARS IN
RELATED QUESTIONS
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 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 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
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 circle with respect to its radius r when r = 5 cm
The total revenue 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 money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) recieved from the sale of x units of a product is given by R(x) = 3x2 + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate ?
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 2 metres high walks at a uniform speed of 5 km/hr away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.
Find an angle θ which increases twice as fast as its cosine ?
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.
Find the point on the curve y2 = 8x for which the abscissa and ordinate change at the same rate ?
The amount of pollution content added in air in a city due to x diesel vehicles is given by P(x) = 0.005x3 + 0.02x2 + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions ?
If \[V = \frac{4}{3}\pi r^3\] , at what rate in cubic units is V increasing when r = 10 and \[\frac{dr}{dt} = 0 . 01\] ? _________________
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 cone whose height is always equal to its diameter is increasing in volume at the rate of 40 cm3/sec. At what rate is the radius increasing when its circular base area is 1 m2?
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
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 the rate of 4π cm3/sec. The rate of increase of the radius when the volume is 288 π cm3, is
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
Each side of an equilateral triangle is increasing at the rate of 8 cm/hr. The rate of increase of its area when side is 2 cm, is
If s = t3 − 4t2 + 5 describes the motion of a particle, then its velocity when the acceleration vanishes, is
Find the rate of change of the area of a circle with respect to its radius r when r = 4 cm.
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?
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`
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?
x and y are the sides of two squares such that y = x – x2. Find the rate of change of the area of second square with respect to the area of first square.
If the rate of change of volume of a sphere is equal to the rate of change of its radius then the surface area of a sphere 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 ______.
A particle moves along the curve 3y = ax3 + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.
If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.
If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?
The median of an equilateral triangle is increasing at the ratio of `2sqrt(3)` cm/s. Find the rate at which its side is increasing.
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?
