A spherical iron ball of radius 10 cm, coated with a layer of ice of uniform thickness, melts at a rate of 100 π cm3/min . The rate at which the thickness of decreases when the thickness of ice is 5 cm, is

Question

A spherical iron ball of radius 10 cm, coated with a layer of ice of uniform thickness, melts at a rate of 100 π cm3/min . The rate at which the thickness of decreases when the thickness of ice is 5 cm, is (A) 1 cm/min (B) 2 cm/min (C)  (1/376)cm/min  (D) 5 cm/min

Tardigrade Admin · Accepted Answer

Given, (dV/dt)=100π cm3/ min ∴ (d/dt)( (4/3)π r3 )=100π ⇒ 3r2(dr/dt)=(300π /4π ) ⇒ ( (dr/dt) )(r=5)=(300/4× 3× 25) =1 text cm/min

Q. A spherical iron ball of radius $10 c m$ , coated with a layer of ice of uniform thickness, melts at a rate of $100 π c m^{3} / min$ . The rate at which the thickness of decreases when the thickness of ice is $5 c m$ , is

$1 c m / min$

$2 c m / min$

$\frac{1}{376} c m / min$

$5 c m / min$

$3 c m / min$

Given, $\frac{d V}{d t} = 100 π c m^{3} / min$
$∴$ $\frac{d}{d t} (\frac{4}{3} π r^{3}) = 100 π$
$\Rightarrow$ $3 r^{2} \frac{d r}{d t} = \frac{300 π}{4 π}$
$\Rightarrow$ $(\frac{d r}{d t})_{(r = 5)} = \frac{300}{4 \times 3 \times 25}$
$= 1 c m / min$

Q. A spherical iron ball of radius 10cm, coated with a layer of ice of uniform thickness, melts at a rate of 100πcm3/min . The rate at which the thickness of decreases when the thickness of ice is 5cm, is

1cm/min

2cm/min

3761​cm/min

5cm/min

3cm/min

Given, dtdV​=100πcm3/min ∴ dtd​(34​πr3)=100π ⇒ 3r2dtdr​=4π300π​ ⇒ (dtdr​)(r=5)​=4×3×25300​ =1cm/min