Question

# An edge of a variable cube is increasing at a rate of $10cm/sec$. How fast the volume of the cube will increase when the edge is $5cm$ long.

A

$750c{m}^{3}/sec$

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B

$75c{m}^{3}/sec$

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C

$300c{m}^{3}/sec$

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D

$150c{m}^{3}/sec$

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E

$25c{m}^{3}/sec$

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Solution

## The correct option is A $750c{m}^{3}/sec$Step 1. Given data:The variable edge of the cube, $\frac{dx}{dt}=10cm/sec$The edge of the cube, $x=5cm$Step 2. Formula used:The volume of the cube, $V={x}^{3}.....\left(1\right)$Step 3. Finding the increase in the volume of the cube:Differentiating both sides of the equation $\left(1\right)$, we get $\frac{dV}{dt}=3{x}^{2}\left(\frac{dx}{dt}\right)\phantom{\rule{0ex}{0ex}}\frac{dV}{dt}=3{\left(5\right)}^{2}\left(10\right)\phantom{\rule{0ex}{0ex}}\frac{dV}{dt}=750c{m}^{3}/sec$Hence, the volume of the cube is increasing at the rate of $750c{m}^{3}/sec$Hence, option A is correct.

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