Question

The edge of a cube was found to be 15 cm with a possible error in measurement of 0.2 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing the volume of the cube and the surface area of the cube. (a) the volume of the cube maximum possible error (in cm3) cm3 relative error (rounded to four decimal places) percentage error (rounded to two decimal places) % (b) the surface area of the cube maximum possible error (in cm2) cm2 relative error (rounded to four decimal places) percentage error (rounded to two decimal places) %

Answer 1

`(a)`

`Maximum\ Error = 27`

`Relative\ Error = 0.008`

`Percentage\ Error = 0.8\%`

`(b)`

`Maximum\ Error = 36`

`Relative\ Error = 0.0267`

`Percentage\ Error = 2.67\%`

Explanation:

Given

`x = 15` -- edge length

`\triangle x = 0.2` -- possible error

Solving (a): Errors in volume

The volume of a cube is:

`V(x) = x^3`

Differentiate

`(dV)/(dx) = 3x^2`

Make dV the subject

`dV =3x^2dx`

Rewrite as:

`\triangle V = 3x^2 \triangle x`

Substitute values for x and
`\triangle x`

`\triangle V = 3 * 15^2 * 0.2^2`

`\triangle V = 27`

Hence:

`Maximum\ Error = 27`

The relative error is then calculated as:

`Relative\ Error = (Maximum\ Error)/(Volume)`

`Relative\ Error = (27)/(x^3)`

`Relative\ Error = (27)/(15^3)`

`Relative\ Error = (27)/(3375)`

`Relative\ Error = 0.008`

The percentage error is:

`Percentage\ Error = Relative\ Error * 100\%`

`Percentage\ Error = 0.008 * 100\%`

`Percentage\ Error = 0.8\%`

Solving (a): Errors in Surface Area

The volume of a cube is:

`A(x) =6x^2`

Differentiate

`(dA)/(dx) =12x`

Make dA the subject

`dA = 12xdx`

Rewrite as:

`\triangle A =12x\triangle x`

Substitute values for x and
`\triangle x`

`\triangle A =12 *15 * 0.2`

`\triangle A =36`

Hence:

`Maximum\ Error = 36`

The relative error is then calculated as:

`Relative\ Error = (Maximum\ Error)/(Surface\ Area)`

`Relative\ Error = (36)/(6x^2)`

`Relative\ Error = (36)/(6*15^2)`

`Relative\ Error = (36)/(1350)`

`Relative\ Error = 0.0267`

The percentage error is:

`Percentage\ Error = Relative\ Error * 100\%`

`Percentage\ Error = 0.0267* 100\%`

`Percentage\ Error = 2.67\%`