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Q3:
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.
Solution :
Given Variables & Initial Setup
Let $r$ represent the radius of the circle at any time $t$, and $A$ represent the area of the circle. We are provided with the following information:
- Rate of change of the radius with respect to time: $\frac{dr}{dt} = 3 \text{ cm/s}$
- Specific radius at which the rate of change of area is required: $r = 10 \text{ cm}$
Step 1: Establishing the Geometric Relationship
The area $A$ of a circle in terms of its radius $r$ is given by the formula [Per the standard formula for the area of a circle]:
$A = \pi r^2$
Step 2: Differentiating with respect to Time ($t$)
To find the rate of change of the area with respect to time, we apply the Chain Rule of calculus [By the Chain Rule: $\frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}$]:
$\frac{d}{dt}(A) = \frac{d}{dt}(\pi r^2)$
$\frac{dA}{dt} = \pi \cdot \frac{d}{dr}(r^2) \cdot \frac{dr}{dt}$
$\frac{dA}{dt} = \pi \cdot (2r) \cdot \frac{dr}{dt}$
$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$
Step 3: Calculating the Result at $r = 10 \text{ cm}$
Substitute the given values $r = 10 \text{ cm}$ and $\frac{dr}{dt} = 3 \text{ cm/s}$ into the derived equation:
$\frac{dA}{dt} = 2 \cdot \pi \cdot (10 \text{ cm}) \cdot (3 \text{ cm/s})$
$\frac{dA}{dt} = 60\pi \text{ cm}^2/\text{s}$
Visual Representation of the Expansion
Summary of Calculation
| Variable | Value |
|---|---|
| Radius ($r$) | $10 \text{ cm}$ |
| Rate of Radius Change ($\frac{dr}{dt}$) | $3 \text{ cm/s}$ |
| Rate of Area Change ($\frac{dA}{dt}$) | $60\pi \text{ cm}^2/\text{s}$ |
Final Solution: The area of the circle is increasing at the rate of $60\pi \text{ cm}^2/\text{s}$.
More Questions from Class 12 Mathematics Application of Derivatives EXERCISE 6.1
- Q1(a): Find the rate of change of the area of a circle with respect to its radius $r$ when (a) $r = 3$ cm
- Q1(b): Find the rate of change of the area of a circle with respect to its radius $r$ when (b) $r = 4$ cm
- Q2: The volume of a cube is increasing at the rate of $8$ cm$^3$/s. How fast is the surface area increasing when the length of an edge is $12$ cm?
- Q4: An edge of a variable cube is increasing at the rate of $3$ cm/s. How fast is the volume of the cube increasing when the edge is $10$ cm long?
- Q5: A stone is dropped into a quiet lake and waves move in circles at the speed of $5$ cm/s. At the instant when the radius of the circular wave is $8$ cm, how fast is the enclosed area increasing?
- Q6: The radius of a circle is increasing at the rate of $0.7$ cm/s. What is the rate of increase of its circumference?
- Q7(a): 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 (a) the perimeter, and
- Q7(b): 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 (b) the area of the rectangle.
- Q8: A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15$ cm.
- Q9: A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is $10$ cm.
- Q10: 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 ?
- Q11: A particle moves 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.
- Q12: The radius of an air bubble is increasing at the rate of $\frac{1}{2}$ cm/s. At what rate is the volume of the bubble increasing when the radius is $1$ cm?
- Q13: 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 $x$.
- Q14: Sand is pouring from a pipe at the rate of $12$ cm$^3$/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 $4$ cm?
- Q15: The total cost $C(x)$ in Rupees associated with the production of $x$ units of an item is given by $C(x) = 0.007x^3 – 0.003x^2 + 15x + 4000$. Find the marginal cost when $17$ units are produced.
- Q16: 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 $x = 7$.
- Q17: The rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ cm is
- Q18: The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. The marginal revenue, when $x = 15$ is
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