1 markMCQSection A

The rate of change of the area of a circle with respect to its radius r (in $\text{cm}^2/\text{s}$ ), when $r = 6$ cm, is :

Q: The rate of change of the area of a circle with respect to its radius r (in $\text{cm}^2/\text{s}$), when $r = 6$ cm, is :

The correct option is **B) $12\pi$**. **Working:** - Area of a circle: $A = \pi r^2$. - Rate of change with respect to r: $\frac{dA}{dr} = 2\pi r$. - At $r = 6$ cm: $\frac{dA}{dr} = 2\pi(6) = 12\pi \text{ cm}^2/\text{cm}$.

Q: Why does $\frac{dA}{dr}$ equal the circumference?

Because $A = \pi r^2$ differentiates to $2\pi r$, which is exactly the circumference formula. Geometrically, growing the radius adds a thin ring of length $2\pi r$.

Calculus (Applications of Derivatives)

Rate of Change — application of derivatives

Options

(A)

10\pi

(B)

12\pi

(C)

8\pi

(D)

11\pi

Official Answer

The correct option is B) $12\pi$ .

Working:

Area of a circle: $A = \pi r^2$ .
Rate of change with respect to r: $\frac{dA}{dr} = 2\pi r$ .
At $r = 6$ cm: $\frac{dA}{dr} = 2\pi(6) = 12\pi \text{ cm}^2/\text{cm}$ .

rate of changearea of circleA = pi r^2dA/dr = 2 pi r12 piapplication of derivativesr = 6

Marking Scheme

11 mark: $\frac{dA}{dr} = 2\pi r$ and value $12\pi$ at $r = 6$ (option B).

Hint

Differentiate $A = \pi r^2$ to get $\frac{dA}{dr} = 2\pi r$ , then substitute $r = 6$ .

Quick Oral Answer

Since $A = \pi r^2$ , $\frac{dA}{dr} = 2\pi r$ , and at $r = 6$ this equals $12\pi \text{ cm}^2$ per cm.

Analysis & Explanation

This tests the basic rate-of-change application of derivatives.

Concept:

The instantaneous rate of change of area with respect to radius is $\frac{dA}{dr}$ .
For $A = \pi r^2, \frac{dA}{dr} = 2\pi r$ , which is numerically the circumference.

Why B is correct:

$\frac{dA}{dr} = 2\pi r = 2\pi(6) = 12\pi$ .

Why the distractors are wrong:

A ( $10\pi$ ): from $r = 5$ instead of 6.
C ( $8\pi$ ): from $r = 4$ , or from mis-differentiating.
D ( $11\pi$ ): a non-standard value with no valid derivation.

Common Mistakes

1Using circumference $2\pi r$ as the area formula or vice versa.
2Substituting a wrong radius value (5 or 4) yielding $10\pi$ or $8\pi$ .
3Forgetting the factor 2 from differentiating $r^2$ , giving $\pi r = 6\pi$ .

Interesting Facts

The derivative of a circle's area ( $\pi r^2$ ) with respect to radius equals its circumference ( $2\pi r$ ) — a beautiful geometric fact.

This same idea generalises: the derivative of a sphere's volume ( $\frac{4}{3}\pi r^3$ ) with respect to r is its surface area ( $4\pi r^2$ ).

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Frequently Asked Questions

Why does $\frac{dA}{dr}$ equal the circumference?

Because $A = \pi r^2$ differentiates to $2\pi r$ , which is exactly the circumference formula. Geometrically, growing the radius adds a thin ring of length $2\pi r$ .

What are the units of $\frac{dA}{dr}$ here?

Area is in $\text{cm}^2$ and radius in cm, so $\frac{dA}{dr}$ is in $\text{cm}^2$ /cm, i.e. cm — numerically $12\pi$ when $r = 6$ cm.