(A) (B) (C) (D) SOLUTION: Area of circle (A) = = Therefore, The correct option is option (B) .
Choose the correct answer: The rate of change of the area of a circle with respect to its radius r at r=6 cm is:
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Exercise 6.1
The total revenue in rupees received from the sale of x units of a product is given by R(x)= 13x^2+26x +15 Find the marginal revenue when x=7
Marginal Revenue (MR) = = = Now, when MR = 26 x 7 + 26 = 208 Therefore, the required marginal revenue is ` 208.
The total cost C(x) in rupees associated with the production of units of an item given by C (x)= 0.007x^3-0.003x^2-15x+4000 Find the marginal cost when 17 units are produced.
Marginal cost = = = Now, when MC = = 6.069 – 0.102 + 15 = 20.967 Therefore, required Marginal cost is ` 20.97.
Sand is pouring from a pipe at the rate of 12 cm3/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm?
Let the height and radius of the sand-cone formed at time second be cm and cm respectively. According to question, Volume of cone (V) = = = Now, since ...
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
Given: Diameter of the balloon = Radius of the balloon = Volume of the balloon = = cu. units Rate of change of volume w.r.t. = = = =
The radius of an air bubble is increasing at the rate of cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Let cm be the radius of the air bubble at time According to question, is positive = cm/sec ……….(i) Volume of air bubble = = = Therefore, required rate of increase of volume of air bubble...
A particle moves along the curve Find the points on the curve 6y=x^3+2at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Given: Equation of the curve ……….(i) Let be the required point on curve (i) According to the question, ……….(ii) From eq. (i), [From eq. (ii)] Taking Required point is (4, 11)....
A ladder 5 cm 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?
Let AB be the ladder and C is the junction of wall and ground, AB = 5 m B Let CA = meters, CB = meters According to the equation, increases, decreases and = 2 cm/s In AC2 + BC2 = AB2 [Using...
A balloon, which always remains spherical has a variables radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.
Since, V = = = Therefore, the volume is increasing at the rate of cm3/sec.
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
Let cm be the radius of the spherical balloon at time According to the question, Radius of balloon is increasing at the rate of cm sec.
The length of a rectangle is decreasing at the rate of 5 cm/minute. When = 8 cm and = 6 cm, find the rates of change of (a) the perimeter and (b) the area of the rectangle.
Given: Rate of decrease of length of rectangle is 5 cm/minute. is negative = –5 cm/minute Also, Rate of increase of width of rectangle is 4 cm/minute is positive = 4 cm/minute (a) Let denotes...
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of its circumference?
Let cm be the radius of the circle at time Rate of increase of radius of circle = 0.7 cm/sec is positive and = 0.7 cm/sec Let be the circumference of the circle. Rate of change of...
A stone is dropped into a quite lake and waves move in circles at the rate of 5 cm/sec. At the instant when radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
Let cm be the radius of the circular wave at time Rate of increase of radius of circular wave = 5 cm/sec is positive and = 5 cm/sec Let be the enclosed area of the circular wave. Rate of...
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 if 10 cm long?
Let cm be the edge of variable cube at time Rate of increase of edge = 3 cm/sec is positive and = 3 cm/sec Let be the volume of the cube. Rate of change of volume of cube = = = = cm3/sec...
The radius of the 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.
Leave cm alone the range of the circle at time Pace of increment of span of circle = 3 cm/sec dx/dt is positive and = 3 cm/sec Let y be the space of the circle. Pace of progress of space of circle =...
The volume of a cube is increasing at the rate of 8 cm3/sec. How fast is the surface area increasing when the length of an edge is 12 cm?
Let cm be the edge of the cube. Given: Rate of increase of volume of cube = 8 cm3/sec is positive = 8 ……….(i) Let be the surface area of the cube, i.e., Rate of change of surface area of...
Find the rate of change of the area of a circle with respect to its radius when (a) = 3 cm (b) = 4 cm
et denote the area of the circle of variable radius Area of circle Rate of change of area w.r.t. = (a) When cm, then sq. cm (b) When cm, then sq. cm
In figure, if ∠BAC =90° and AD⊥BC. Then, (a) BD.CD = BC² (b) AB.AC = BC² (c) BD.CD=AD² (d) AB.AC =AD²
Solution: c) BD.CD=AD² Explanation: From triangles ADB and ADC, Now according to the question, we have, ∠ADB = ∠ADC = 90° (Since AD ⊥ BC) ∠DBA = ∠DAC [As each angle = 90°- ∠C] Using AAA criterion...