Question

The percentage error in the volume of a cube is 9%. What is % error in the length of edge of the cube ?

Answer 1

Given that,

Percentage error in the volume of a cube = 9%

Let the edge of the cube is x.

The volume of cube will be x³.

We need to calculate the percentage error in the length of edge of the cube

Using volume of cube

$V=x^3$

On differentiating

$\dfrac{dV}{dx}=3x^2$

For percentage error,

$\Delta V=\dfrac{dV}{dx}\times\Delta x$

Put the value in to the formula

$\Delta V=3x^2\Delta x$

Dividing by V on both side

$\dfrac{\Delta V}{V}=\dfrac{3x^2\Delta x}{V}$

$\dfrac{\Delta V}{V}=\dfrac{3x^2}{x^3}\times\Delta x$

$\dfrac{\Delta V}{V}=3\times\dfrac{\Delta x}{x}$

Put the value into the formula

$9=3\times\dfrac{\Delta x}{x}$

$\dfrac{\Delta x}{x}=\dfrac{9}{3}$

$\dfrac{\Delta x}{x}=3\%$

Hence, The percentage error in the length of edge of the cube is 3%.

Answer 2

Given that,

Percentage error in the volume of a cube = 9%

Let the edge of the cube is x.

The volume of cube will be x³.

We need to calculate the percentage error in the length of edge of the cube

Using volume of cube

$V=x^3$

On differentiating

$\dfrac{dV}{dx}=3x^2$

For percentage error,

$\Delta V=\dfrac{dV}{dx}\times\Delta x$

Put the value in to the formula

$\Delta V=3x^2\Delta x$

Dividing by V on both side

$\dfrac{\Delta V}{V}=\dfrac{3x^2\Delta x}{V}$

$\dfrac{\Delta V}{V}=\dfrac{3x^2}{x^3}\times\Delta x$

$\dfrac{\Delta V}{V}=3\times\dfrac{\Delta x}{x}$

Put the value into the formula

$9=3\times\dfrac{\Delta x}{x}$

$\dfrac{\Delta x}{x}=\dfrac{9}{3}$

$\dfrac{\Delta x}{x}=3\%$

Hence, The percentage error in the length of edge of the cube is 3%.

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