The area of the circle is increasing at a uniform rate of 2 cm²/sec. How fast is the circumference of the circle increasing when the radius r=5 cm ?

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2024: question 23

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, section B, question 23, for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23 · 2 marks · Very short answerApplication of Derivatives

The area of the circle is increasing at a uniform rate of $2 cm^{2} / sec$ . How fast is the circumference of the circle increasing when the radius $r = 5$ cm ?

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Write area and circumference in terms of r and differentiate both with respect to t.

Hint 2

Use $\frac{d A}{d t} = 2$ to find $\frac{d r}{d t}$ first, then substitute into $\frac{d C}{d t}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Area $A = π r^{2} \Rightarrow \frac{d A}{d t} = 2 π r \frac{d r}{d t} = 2 \Rightarrow \frac{d r}{d t} = \frac{1}{π r}$ cm/sec.
1 mark
Circumference $C = 2 π r \Rightarrow \frac{d C}{d t} = 2 π \frac{d r}{d t} = 2 π \cdot \frac{1}{π r} = \frac{2}{r}$ . At $r = 5$ cm, $\frac{d C}{d t} = \frac{2}{5}$ cm/sec.
1 mark

Answer: $\frac{2}{5}$ cm/sec (0.4 cm/sec)

Where the marks are

Differentiates $A = π r^{2}$ and gets $\frac{d r}{d t} = \frac{1}{π r}$ (1 mark)
Gets $\frac{d C}{d t} = \frac{2}{r} = \frac{2}{5}$ cm/sec at $r = 5$ (1 mark)

Revise the chapter

Chapter: Application of Derivatives (notes, formulas, where marks are lost and every past question)
Topic: Rate of Change of Quantities

Practise this chapterBoard-style questions from Application of Derivatives, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 23

Hints

Step-by-step solution

Revise the chapter

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