Questions and Solutions
Question 1:
Find the rate of change of the area of a circle with respect to its radius \( r \) when (a) \( r = 3\;\text{cm} \), (b) \( r = 4\;\text{cm} \).
Question 2:
The volume of a cube is increasing at the rate of \( 8\;\text{cm}^3/\text{s} \). How fast is the surface area increasing when the length of its edge is \( 12\;\text{cm} \)?
Question 3:
An edge of a variable cube is increasing at the rate of \( 3\;\text{cm}/\text{s} \). How fast is the volume of the cube increasing when the edge is \( 10\;\text{cm} \) long?
Question 4:
A stone is dropped into a quiet lake and waves move in circles at the speed of \( 5\;\text{cm}/\text{s} \). At the instant when the radius of the circular wave is \( 8\;\text{cm} \), how fast is the enclosed area increasing?
Question 5:
The radius of a circle is increasing at the rate of \( 0.7\;\text{cm}/\text{s} \). What is the rate of increase of its circumference?
Question 6:
The length \( x \) of a rectangle is decreasing at the rate of \( 5\;\text{cm}/\text{min} \) and the width \( y \) is increasing at the rate of \( 4\;\text{cm}/\text{min} \). When \( x = 8\;\text{cm} \) and \( y = 6\;\text{cm} \), find the rates of change of (a) the perimeter, and (b) the area of the rectangle.
Question 7:
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 \( 15\;\text{cm} \).
Question 8:
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the latter is \( 10\;\text{cm} \).
Question 9:
A ladder \( 5\;\text{m} \) long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of \( 2\;\text{cm}/\text{s} \). How fast is its height on the wall decreasing when the foot of the ladder is \( 4\;\text{m} \) away from the wall?
Question 10:
A particle is moving along the curve \( 6y = x^3 + 2 \). Find the points on the curve at which the \( y \)-coordinate is changing 8 times as fast as the \( x \)-coordinate.
Question 11:
The radius of an air bubble is increasing at the rate of \( \frac{1}{2}\;\text{cm}/\text{s} \). At what rate is the volume of the bubble increasing when the radius is \( 1\;\text{cm} \)?
Question 12:
A balloon, which always remains spherical, has a variable diameter \( \frac{3}{2}(2x + 1) \). Find the rate of change of its volume with respect to \( x \).
Question 13:
Sand is pouring from a pipe at the rate of \( 12\;\text{cm}^3/\text{s} \). 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 \( 4\;\text{cm} \)?
Question 14:
The rate of change of the area of a circle with respect to its radius \( r \) at \( r = 6\;\text{cm} \) is: (A) \( 10\pi \) (B) \( 12\pi \) (C) \( 8\pi \) (D) \( 12\pi \)