A man, $2$ m tall, walks at the rate of $1\tfrac{2}{3}$ m/s towards a street light which is $5\tfrac{1}{3}$ m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of his shadow changing when he is $3\tfrac{1}{3}$ m from the base of the light?

A man, 2 m tall, walks at the rate of 1(2)/(3) m/s towards a street light which is 5(1)/(3) m above the ground

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Concept understanding — Related Rates

Related Rates

The idea: quantities that change together

Many real situations involve two or more quantities that all vary with time, linked by a fixed relationship. Inflate a balloon and its radius and volume both grow; slide a ladder down a wall and the top's height and the foot's distance both change. A related-rates problem gives you the rate at which one quantity is changing and asks for the rate of another, at some instant.

The key insight: if the quantities are tied together by an equation, then their rates are tied together too. We uncover that link by differentiating the equation with respect to time $t$ .

The core mechanism: differentiate with respect to time

Every variable is a function of $t$ , so differentiating brings in the chain rule — each variable's derivative picks up a factor of its own rate. For example, if the volume of a sphere is $V=\tfrac{4}{3}\pi r^{3}$ , then differentiating both sides with respect to $t$ gives

\frac{dV}{dt} = 4\pi r^{2}\,\frac{dr}{dt}.

This single equation connects the rate the volume grows, $\dfrac{dV}{dt}$ , to the rate the radius grows, $\dfrac{dr}{dt}$ . Knowing one (and the current $r$ ) gives the other.

The standard procedure

t

Worked example

Air is pumped into a spherical balloon at $\dfrac{dV}{dt}=100\ \text{cm}^3/\text{s}$ . How fast is the radius increasing when $r=5\ \text{cm}$ ?

From $\dfrac{dV}{dt}=4\pi r^{2}\dfrac{dr}{dt}$ , substitute $\dfrac{dV}{dt}=100$ and $r=5$ :

$100 = 4\pi(5)^2\,\frac{dr}{dt} = 100\pi\,\frac{dr}{dt} \implies \frac{dr}{dt} = \frac{1}{\pi}\ \text{cm/s}.$ …

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With shadow length $s=\frac{3}{5}x$ , the shadow tip moves toward the light at $\frac{8}{3}$ m/s and the shadow length decreases at $1$ m/s — both rates constant, independent of the $3\tfrac{1}{3}$ m distance.

The intuition

A man walking toward a street light casts a shadow that shrinks. The lamp-top, the man's head, and the tips of the two shadows form two similar right triangles: a big one from the lamp to the shadow tip, and a small one from the man's head to the same tip. Similar triangles give a fixed relation between the shadow length and the man's distance, and differentiating in time turns "how fast he walks" into "how fast the shadow changes."

Set up with similar triangles

Let $x$ = the man's distance from the base of the light and $s$ = the length of his shadow. The lamp is $H=5\tfrac{1}{3}=\frac{16}{3}$ m high and the man is $h=2$ m tall. The large and small triangles share the shadow tip, so

\frac{H}{x+s}=\frac{h}{s}\ \Longrightarrow\ \frac{16/3}{x+s}=\frac{2}{s}.

Work the steps

1. Relate $s$ and $x$ .

\frac{16}{3}s=2(x+s)\ \Longrightarrow\ \left(\frac{16}{3}-2\right)s=2x\ \Longrightarrow\ \frac{10}{3}s=2x\ \Longrightarrow\ s=\frac{3}{5}x.

2. The given rate. He moves toward the light, so $x$ decreases:

\frac{dx}{dt}=-\frac{5}{3}\ \text{m/s}.

3. Rate of change of the shadow length.

$\frac{ds}{dt}=\frac{3}{5}\frac{dx}{dt}=\frac{3}{5}\left(-\frac{5}{3}\right)=-1\ \text{m/s}.$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 15 on this concept.

CBSE 2025Set 65/2/11 markMCQ
$100\pi$
›Reveal solutionSolution
The volume of a cylinder is $V = \pi r^2 h$ . Since the radius is constant, the rate of change of volume with respect to time is $\frac{dV}{dt} = \pi r^2 \frac{dh}{dt}$ . Given $\frac{dV}{dt} = 100\pi$ cm³/s and $r = 10$ cm, solving gives $\frac{dh}{dt} = 1$ cm/s. The correct option is (C).

This is a classic Related Rates problem. The core idea is that when two quantities are linked by a geometric formula (here, volume and height of a cylinder), their rates of change with respect to time are also linked. You differentiate the relationship with respect to time, plug in what you know, and solve for the unknown rate.

The key insight: the tank’s radius is fixed at 10 cm. So as sugar pours in, the height increases, but the cross-sectional area stays the same. That means the volume increases at a constant rate per unit height — specifically, each 1 cm rise in height adds $\pi (10)^2 = 100\pi$ cm³ of volume. Since sugar is being added at exactly $100\pi$ cm³/s, the height must be rising at 1 cm/s.

Let’s work it out formally.

Write the relationship between volume and height. For a cylinder, $V = \pi r^2 h$ . Here $r = 10$ cm, so

$V = \pi (10)^2 h = 100\pi h.$

Differentiate both sides with respect to time $t$ .

Since $r$ is constant, $\frac{dV}{dt} = 100\pi \frac{dh}{dt}$ .

This is the related rates equation — it tells us how fast the volume changes in terms of how fast the height changes.

Substitute the given rate.

We know $\frac{dV}{dt} = 100\pi$ cm³/s. So:

$100\pi = 100\pi \frac{dh}{dt}.$

Solve for $\frac{dh}{dt}$ . Divide both sides by $100\pi$ : …
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CBSE 2026Set ANNUAL1 mark
$Q. The edge of a variable cube is increasing at the rate of 3 cm/s. The volume of the cube is increasing at the rate of __________ while the edge is 10 cm long.$
›Reveal solutionSolution

Use $V=e^3$ and the chain rule $dV/dt = 3e^2\,de/dt$ .

Let $e$ be the edge; $V=e^3$ , so $\dfrac{dV}{dt}=3e^2\dfrac{de}{dt}$ .

Given $\dfrac{de}{dt}=3$ cm/s and $e=10$ cm:

…

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CBSE 2026Set ANNUAL1 mark
$1/2$
›Reveal solutionSolution

Use $V=\dfrac43\pi r^3$ and $\dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$ .

Given $\dfrac{dr}{dt}=\dfrac12$ cm/s, at $r=1$ cm:

…

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CBSE 2026Set ANNUAL1 markMCQ
$Q. Radius of a circle is increasing at the rate of 1/π m/s. Rate of change of its circumference is:$
(a) 4π m/s
(b) 2 m/s
(c) 2π m/s
(d) 4 m/s
›Reveal solutionSolution

Since $C = 2\pi r$ , differentiating both sides w.r.t. time gives $\frac{dC}{dt} = 2\pi \frac{dr}{dt}$ directly.

The circumference of a circle of radius $r$ is $C = 2\pi r$ .

Differentiating with respect to time $t$ :

$\frac{dC}{dt} = 2\pi \frac{dr}{dt}$

Given $\frac{dr}{dt} = \frac{1}{\pi}$ m/s, substitute: …

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CBSE 2025Set ANNUAL1 markMCQ
$Q. Radius of a circle is increasing at the rate of 2 m/s. Rate of change of its circumference is:$
(a) 4π m/s
(b) 2 m/s
(c) 2π m/s
(d) 4 m/s
›Reveal solutionSolution

Differentiate the circumference formula $C=2\pi r$ with respect to time and plug in $\dfrac{dr}{dt}$ .

Given $\dfrac{dr}{dt} = 2$ m/s. Circumference $C = 2\pi r$ .

Differentiating both sides with respect to time $t$ : …

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CBSE 2024Set ANNUAL1 mark
$Q. The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.$
›Reveal solutionSolution

Use related rates: differentiate $A=\pi r^2$ w.r.t. time and substitute the given $\dfrac{dr}{dt}$ and $r$ .

Given $\dfrac{dr}{dt} = 3$ cm/s, find $\dfrac{dA}{dt}$ at $r = 10$ cm.

$A = \pi r^2 \Rightarrow \dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt}$

…

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CBSE 2024Set ANNUAL1 mark
$0.7$
›Reveal solutionSolution

Differentiate the circumference formula $C=2\pi r$ with respect to time.

The circumference of a circle of radius $r$ is

$C = 2\pi r.$

Given $\dfrac{dr}{dt} = 0.7$ cm/s. Differentiating with respect to $t$ : …

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CBSE 2023Set ANNUAL1 markMCQ
$0.3$
(a) $0.4\pi$ cm/sec
(b) $0.6\pi$ cm/sec
(c) $0.8\pi$ cm/sec
(d) none of these
›Reveal solutionSolution

This is a related-rates problem: differentiate the perimeter formula w.r.t. time.

Perimeter (circumference) $P = 2\pi r$ . Differentiating w.r.t. time $t$ : $\dfrac{dP}{dt} = 2\pi\dfrac{dr}{dt}$ .

…

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CBSE 2023Set ANNUAL1 mark
$3$
›Reveal solutionSolution

Differentiate $A=\pi r^2$ with respect to time and substitute $r=10$ , $dr/dt=3$ .

Area of circle: $A=\pi r^2$ . Differentiating both sides with respect to $t$ :

$\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}$

…

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CBSE 2023Set ANNUAL1 markMCQ
$Q. The radius of a circle is increasing at the rate of 0.7 cm/s. The rate of increase of its circumference is -$
(a) 7.1 cm/s
(b) 4.0 cm/s
(c) 3.9 cm/s
(d) 4.4 cm/s
›Reveal solutionSolution

Differentiate $C = 2\pi r$ with respect to time and substitute the given rate of change of the radius.

Circumference $C = 2\pi r$ . Differentiating with respect to time $t$ :

$\frac{dC}{dt} = 2\pi\frac{dr}{dt}.$

Given $\dfrac{dr}{dt} = 0.7$ cm/s: …

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CBSE 2018Set ANNUAL1 markMCQ
$cm^3$
(a) 5 $cm^2$ /sec.
(b) 10 $cm^2$ /sec.
(c) 15 $cm^2$ /sec.
(d) 20 $cm^2$ /sec.
›Reveal solutionSolution

related rates: relate dV/dt to dr/dt, then dS/dt to dr/dt

Volume $V=\dfrac43\pi r^3\Rightarrow \dfrac{dV}{dt}=4\pi r^2\dfrac{dr}{dt}$ .

At $r=5$ : $25=4\pi(25)\dfrac{dr}{dt}\implies \dfrac{dr}{dt}=\dfrac{25}{100\pi}=\dfrac{1}{4\pi}$ .

Surface area $S=4\pi r^2\Rightarrow \dfrac{dS}{dt}=8\pi r\dfrac{dr}{dt}$ .

…

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CBSE 2018Set ANNUAL1 markMCQ
$x$
(a) $\dfrac{\pi}{3}$
(b) $\dfrac{\pi}{2}$
(c) $\pi$
(d) $\dfrac{3\pi}{2}$
›Reveal solutionSolution

Translate "x increases twice as fast as sin x" into $dx/dt = 2\,d(\sin x)/dt$ and solve for x.

"x increases twice as fast as its sine" means $\dfrac{dx}{dt} = 2\dfrac{d(\sin x)}{dt} = 2\cos x\dfrac{dx}{dt}$ …

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Q.A man, $2$ m tall, walks at the rate of $1\tfrac{2}{3}$ m/s towards a street light which is $5\tfrac{1}{3}$ m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of his shadow changing when he is $3\tfrac{1}{3}$ m from the base of the light?

Concept understanding — Related Rates

Related Rates

The idea: quantities that change together

The core mechanism: differentiate with respect to time

The standard procedure

Worked example

The intuition

Set up with similar triangles

Work the steps

Method: Related Rates via Similar Triangles (Shadow / Lamp-Post Problems)

Steps

Common Mistakes

The intuition

Set up with similar triangles

Work the steps

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