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Solved Examples
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 cc  The rate of change of the circumference C of a circle with respect to its area A is k πA. Find the value of 6k² + 3k.  cc View Solution
 cc  The side length of an equilateral triangle is l cm. If the rate of change of area of the incircle with respect to l is kπl6, then find the value of 6k + 49.  cc View Solution
 cc  Find the value(s) of x for which the rate of change of y = x33 + 8x2 + 64x with respect to x is 49.  cc View Solution
 cc  A small stone is dropped into a quiet pond and circular ripples spread over the surface of water. The radius of each of these ripples increase at the rate of 20 inches per second. Find the rate at which the area inside the circle is increasing at the instant the radius is 5 ft.  cc View Solution
 cc  x = 4 cos 5θ and y = 4 sin 5θ, where 0 ≤ θ ≤ π. Find the value of θ at which the rate of change of x and y with respect to θ are equal.  cc View Solution
 cc  The rate of change of volume of a cube with respect to its length x is equal to 11k times its total surface area A. The value of k is  cc View Solution
 cc  An airplane at an altitude of 900 m flying horizontally with a speed of 250 m/sec passes directly over an observer. Find the rate at which the plane approaches the observer, when it is at 904 m away from the observer.  cc View Solution
 cc  If the rate of change of volume V of a sphere with respect to its surface area S is k4Sπ, then find the value of (33k + 8k²).  cc View Solution
 cc  The rate of change of each side length of an equilateral triangle is 5 cm/s. Find the corresponding rate of change in its area when the side lengths are 8 cm each.  cc View Solution
 cc  At a given instant, the legs of a right triangle are 9 cm and 9 cm respectively. The length of the first leg increases at the rate of 2 cm/s and the length of the second leg increases at the rate of 1cm/s. What is the rate of change in the area of the triangle after 4 sec?  cc View Solution
 cc  An equilateral triangle is inscribed in a circle. The area of the circle is decreasing at the rate of 4π cm²/s. Determine the rate of change in the side length of the equilateral triangle at the instant its side length is 5 cm.  cc View Solution
 cc  The base radius of a right circular cylindrical oil container is 90 m. Oil is drawn from it at the rate of 729000 m3 per minute. Find the rate at which the oil level falls in the container.  cc View Solution
 cc  A gas balloon contains 1200 cubic ft. of gas at a pressure of 60 lb per ft². The pressure is decreasing at the rate of 0.1 lb per ft². per second. Find the rate at which the volume increases, if the gas obeys Boyle's law, PV = constant.  cc View Solution
 cc  A balloon is in the shape of a cone surmounted by a hemisphere of same radius as that of the base of the cone. The height of the cone is always equal to the diameter of the hemisphere. The total height and volume of the balloon are h and V. The rate of change of V is 34k times the rate of change of h. Find the value of k.  cc View Solution
 cc  Sand is being poured at the rate of 7000 cm3/son the ground from the orifice of an elevated pipe and forms a pile, which has always the shape of a right circular cone whose height is equal to the base radius. Find the rate at which the height of the pile is rising when the height is 50 cm.  cc View Solution
 cc  The length of the diagonal of a cube is increasing at the rate of 4 cm/s. What is the rate of change in the surface area of the cube when its side length measures 2 cm?  cc View Solution
 cc  The height of a right circular cone is decreasing at the rate of 2 cm/s and the radius of its base is increasing at the rate of 3 cm/s. Find the rate of change of volume of the cone when the height is 2 cm and the base radius is 3 cm.  cc View Solution
 cc  A rod of length 18 m moves in such a way that its ends A and B always touches the X and Y axes. If A is at 9 m from the origin and is moving away at the rate of 0.9 m/s, then find the rate at which the area of the triangle formed by the rod with the axes is decreasing.  cc View Solution
 cc  A square is inscribed in a circle. If the side length of the square is increasing at the rate of 3 cm/s, then what is the rate of increase of its area when its side length measures 3 cm?  cc View Solution
 cc  A conical vessel of height 10 m and radius 5 m is being filled with water at a uniform rate of 6 cubic m per minute. Find the rate at which the level of water in the conical vessel is rising when the depth is 5 m.  cc View Solution
 cc  A variable triangle ABC is inscribed in a circle of diameter x units. At a particular instant, the rate of change of side BC is x4 times the rate of change of measure of the opposite angle A. Find the measure of angle A.  cc View Solution
 cc  If the semi vertical angle of a right circular cone is 45o and the rate of change of volume of the cone is πrkdrdt, then find the value of (k + 2)(k + 4)(k + 5).  cc View Solution
 cc  The side length of a square increases at the rate of 2 cm/s. At what rate is the area of the square increasing when its side length is 20 cm?  cc View Solution
 cc  The volume of a metallic hollow sphere is constant. Its outer radius is increasing at the rate of 2 cm/s. Find the rate at which its inner radius is increasing if the outer and inner radius of the sphere are 10 cm and 5 cm respectively.  cc View Solution
 cc  The rate of change of each side length of an equilateral triangle is 4 cm/s. Find the corresponding rate of change in its area when the side lengths are 12 cm each.  cc View Solution
 cc  At a given instant, the legs of a right triangle are 6 cm and 7 cm respectively. The length of the first leg increases at the rate of 2 cm/s and the length of the second leg increases at the rate of 1cm/s. What is the rate of change in the area of the triangle after 3 sec?  cc View Solution
 cc  The volume of a metallic hollow sphere is constant. Its outer radius is increasing at the rate of 3 cm/s. Find the rate at which its inner radius is increasing if the outer and inner radius of the sphere are 8 cm and 4 cm respectively.  cc View Solution
 cc  An equilateral triangle is inscribed in a circle. The area of the circle is decreasing at the rate of 3π cm²/s. Determine the rate of change in the side length of the equilateral triangle at the instant its side length is 6 cm.  cc View Solution
 cc  An airplane at an altitude of 500 m flying horizontally with a speed of 250 m/sec passes directly over an observer. Find the rate at which the plane approaches the observer, when it is at 525 m away from the observer.  cc View Solution
 cc  Find the value(s) of x for which the rate of change of y = x33 + 6x2 + 36x with respect to x is 25.  cc View Solution
 cc  If the rate of change of volume V of a sphere with respect to its surface area S is k4Sπ, then find the value of (14k + 7k²).  cc View Solution
 cc  A gas balloon contains 800 cubic ft. of gas at a pressure of 40 lb per ft². The pressure is decreasing at the rate of 0.1 lb per ft². per second. Find the rate at which the volume increases, if the gas obeys Boyle's law, PV = constant.  cc View Solution
 cc  If the semi vertical angle of a right circular cone is 45o and the rate of change of volume of the cone is πrkdrdt, then find the value of (k + 2)(k + 3)(k + 3).  cc View Solution
 cc  The base radius of a right circular cylindrical oil container is 30 m. Oil is drawn from it at the rate of 27000 m3 per minute. Find the rate at which the oil level falls in the container.  cc View Solution
 cc  The rate of change of the circumference C of a circle with respect to its area A is k πA. Find the value of 7k² + 5k.  cc View Solution
 cc  The rate of change of volume of a cube with respect to its length x is equal to 37k times its total surface area A. The value of k is  cc View Solution
 cc  x = 4 cos 9θ and y = 4 sin 9θ, where 0 ≤ θ ≤ π. Find the value of θ at which the rate of change of x and y with respect to θ are equal.  cc View Solution
 cc  A small stone is dropped into a quiet pond and circular ripples spread over the surface of water. The radius of each of these ripples increase at the rate of 16 inches per second. Find the rate at which the area inside the circle is increasing at the instant the radius is 4 ft.  cc View Solution
 cc  The side length of a square increases at the rate of 7 cm/s. At what rate is the area of the square increasing when its side length is 70 cm?  cc View Solution
 cc  A man 4 ft tall is walking away from a lamp-post of height 18 ft at the rate of 12 ft/min. Find the rate at which the length of his shadow increases.  cc View Solution
 cc  The side length of an equilateral triangle is l cm. If the rate of change of area of the incircle with respect to l is kπl6, then find the value of 8k + 43.  cc View Solution
 cc  A variable triangle ABC is inscribed in a circle of diameter x units. At a particular instant, the rate of change of side BC is x6 times the rate of change of measure of the opposite angle A. Find the measure of angle A.  cc View Solution
 cc  A balloon is in the shape of a cone surmounted by a hemisphere of same radius as that of the base of the cone. The height of the cone is always equal to the diameter of the hemisphere. The total height and volume of the balloon are h and V. The rate of change of V is 44k times the rate of change of h. Find the value of k.  cc View Solution
 cc  Sand is being poured at the rate of 6000 cm3/son the ground from the orifice of an elevated pipe and forms a pile, which has always the shape of a right circular cone whose height is equal to the base radius. Find the rate at which the height of the pile is rising when the height is 30 cm.  cc View Solution
 cc  The length of the diagonal of a cube is increasing at the rate of 2 cm/s. What is the rate of change in the surface area of the cube when its side length measures 5 cm?  cc View Solution
 cc  The height of a right circular cone is decreasing at the rate of 3 cm/s and the radius of its base is increasing at the rate of 2 cm/s. Find the rate of change of volume of the cone when the height is 3 cm and the base radius is 2 cm.  cc View Solution
 cc  A rod of length 4 m moves in such a way that its ends A and B always touches the X and Y axes. If A is at 2 m from the origin and is moving away at the rate of 0.2 m/s, then find the rate at which the area of the triangle formed by the rod with the axes is decreasing.  cc View Solution
 cc  A square is inscribed in a circle. If the side length of the square is increasing at the rate of 3 cm/s, then what is the rate of increase of its area when its side length measures 8 cm?  cc View Solution
 cc  A conical vessel of height 12 m and radius 6 m is being filled with water at a uniform rate of 7 cubic m per minute. Find the rate at which the level of water in the conical vessel is rising when the depth is 6 m.  cc View Solution