Question

The volume of a cube is increasing at the rate
of 6 cm^3/min. How fast in the surface are increasing
When the length of an edge is 12cm?​

Answer 1

Given :

• The volume of a cube is increasing at the rate of 6 cm³/min.

• length of an edge = 12cm

To find :

• How fast in the surface area increasing

Formula used :

• V= a³

• S= 6 a²

where :-

• V = Volume of cube

• S = Surface area of cube

Solution :

The volume of a cube is increasing at the rate of 6 cm³/min.

Which means dv/dt = 6 cm³/min

$\implies \: V = {a}^{3}$

$\implies \: \dfrac{dV}{dt} = \dfrac{ d({a}^{3}) }{dt}$

$\implies \: \dfrac{dV}{dt} =3 {a}^{2} \dfrac{ d({a}) }{dt}$

Now put :-

• dv/dt = 6
• a = 12

$\implies \: 6 =3 \times {12}^{2} \times \dfrac{ d({a}) }{dt}$

$\implies \: 6 =3 \times {12} \times 12 \times \dfrac{ d({a}) }{dt}$

$\implies \: \dfrac{6}{3 \times {12} \times 12 } = \dfrac{ d({a}) }{dt}$

$\implies \: \dfrac{1}{3 \times {2} \times 12 } = \dfrac{ d({a}) }{dt}$

$\implies \: \dfrac{1}{72 } = \dfrac{ d({a}) }{dt}$

Now :-

$S = 6 {a}^{2}$

$\implies\dfrac{dS}{dt} = \dfrac{d( 6 {a}^{2}) }{dt}$

$\implies\dfrac{dS}{dt} = \dfrac{6 \times d( {a}^{2}) }{dt}$

$\implies\dfrac{dS}{dt} = 12a \: \dfrac{d(a)}{dt}$

Now put :-

• da/dt = 1/72
• a = 12

$\implies\dfrac{dS}{dt} = 12 \times 12 \times \dfrac{1}{72}$

Therefore, ds/dt is rate at which surface area is increasing.

$\implies\dfrac{dS}{dt} = \dfrac{144}{72}$

$\implies\dfrac{dS}{dt} = 2$

Answer :

Rate at which surface area increasing = 2 cm²/min.

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$\large\bf\blue{Additional-Information}$

• Volume of cylinder = πr²h

• T.S.A of cylinder = 2πrh + 2πr²

• Volume of cone = ⅓ πr²h

• C.S.A of cone = πrl

• T.S.A of cone = πrl + πr²

• Volume of cuboid = l × b × h

• C.S.A of cuboid = 2(l + b)h

• T.S.A of cuboid = 2(lb + bh + lh)

• C.S.A of cube = 4a²

• T.S.A of cube = 6a²

• Volume of cube = a³

• Volume of sphere = (4/3)πr³

• Surface area of sphere = 4πr²

• Volume of hemisphere = (⅔) πr³

• C.S.A of hemisphere = 2πr²

• T.S.A of hemisphere = 3πr²

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• d(sinx)/dx = cosx

• d(cos x)/dx = -sin x

• d(cosec x)/dx = -cot x cosec x

• d(tan x)/dx = sec²x

• d(sec x)/dx = secx tanx

• d(cot x)/dx = - cosec² x

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