JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Application of Derivatives

WhatsApp Email

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is :

(A)9✓ Correct
(B)10
(C)11
(D)12

Answer: (A)

Solution

Let

r

be the radius and

S = 4 π r^{2}

the surface area. Given

\frac{d S}{d t} = k

(constant), we have

8 π r \frac{d r}{d t} = k

, so

4 π r^{2} = k t + C

. At

t = 0

r = 3

36 π = C

. At

t = 5

r = 7

196 π = 5 k + 36 π

, giving

k = 32 π

. Thus

4 π r^{2} = 32 π t + 36 π

, or

r^{2} = 8 t + 9

. At

t = 9

r^{2} = 72 + 9 = 81

, so

r = 9

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Single correct option

Practise more: all JEE Main PYQs on Application of Derivatives →

More JEE Main questions on Application of Derivatives

Let $f : R \to R$ be a twice differentiable function such that the quadratic equation $f (x) m^{2} - 2 f^{'} (x) m + f^{'} (x) = 0$ in m , has two equal roots for every …
JEE Main 2026View solution →
Let $f : R \to R$ be a twice differentiable function such that $f^{''} (x) > 0$ for all $x \in R$ and $f^{'} (a - 1) = 0$ , where a is real number. Let $g (x) = f (tan^{2} x - 2 tan x + a)$ , $0 < x < \frac{π}{2}$ .

Consider the following two statements:
(I) g i …
JEE Main 2026View solution →
Let the maximum value of $(sin^{- 1} x)^{2} + (cos^{- 1} x)^{2}$ for $x \in [- \frac{3}{2}, \frac{1}{2}]$ be $\frac{m}{n} π^{2}$ , where $gcd (m, n) = 1$ .

Then $m + n$ is equal to ____.
JEE Main 2026View solution →
Let $f (x) = x^{2025} - x^{2000}, x \in [0, 1]$ and the minimum value of the function $f (x)$ in the interval $[0, 1]$ be $(80)^{80} (n)^{- 81}$ . Then n is equal to
JEE Main 2026View solution →
Let $(2 α, α)$ be the largest interval in which the function $f (t) = \frac{∣ t + 1∣}{t ^{2}}, t < 0$ , is strictly decreasing. Then the local maximum value of the function
$g (x) = 2 lo g_{e} (x - 2) + α x^{2} + 4 x - α, x > 2$ , is ______
JEE Main 2026View solution →
Let f be a differentiable function satisfying
$f (x) = 1 - 2 x + \int_{0}^{x} e^{(x - t)} f (t) dt, x \in R$ and let …
JEE Main 2026View solution →