Answer:
[tex](a)[/tex]
[tex]Maximum\ Error = 27[/tex]
[tex]Relative\ Error = 0.008[/tex]
[tex]Percentage\ Error = 0.8\%[/tex]
[tex](b)[/tex]
[tex]Maximum\ Error = 36[/tex]
[tex]Relative\ Error = 0.0267[/tex]
[tex]Percentage\ Error = 2.67\%[/tex]
Step-by-step explanation:
Given
[tex]x = 15[/tex] -- edge length
[tex]\triangle x = 0.2[/tex] -- possible error
Solving (a): Errors in volume
The volume of a cube is:
[tex]V(x) = x^3[/tex]
Differentiate
[tex]\frac{dV}{dx} = 3x^2[/tex]
Make dV the subject
[tex]dV =3x^2dx[/tex]
Rewrite as:
[tex]\triangle V = 3x^2 \triangle x[/tex]
Substitute values for x and [tex]\triangle x[/tex]
[tex]\triangle V = 3 * 15^2 * 0.2^2[/tex]
[tex]\triangle V = 27[/tex]
Hence:
[tex]Maximum\ Error = 27[/tex]
The relative error is then calculated as:
[tex]Relative\ Error = \frac{Maximum\ Error}{Volume}[/tex]
[tex]Relative\ Error = \frac{27}{x^3}[/tex]
[tex]Relative\ Error = \frac{27}{15^3}[/tex]
[tex]Relative\ Error = \frac{27}{3375}[/tex]
[tex]Relative\ Error = 0.008[/tex]
The percentage error is:
[tex]Percentage\ Error = Relative\ Error * 100\%[/tex]
[tex]Percentage\ Error = 0.008 * 100\%[/tex]
[tex]Percentage\ Error = 0.8\%[/tex]
Solving (a): Errors in Surface Area
The volume of a cube is:
[tex]A(x) =6x^2[/tex]
Differentiate
[tex]\frac{dA}{dx} =12x[/tex]
Make dA the subject
[tex]dA = 12xdx[/tex]
Rewrite as:
[tex]\triangle A =12x\triangle x[/tex]
Substitute values for x and [tex]\triangle x[/tex]
[tex]\triangle A =12 *15 * 0.2[/tex]
[tex]\triangle A =36[/tex]
Hence:
[tex]Maximum\ Error = 36[/tex]
The relative error is then calculated as:
[tex]Relative\ Error = \frac{Maximum\ Error}{Surface\ Area}[/tex]
[tex]Relative\ Error = \frac{36}{6x^2}[/tex]
[tex]Relative\ Error = \frac{36}{6*15^2}[/tex]
[tex]Relative\ Error = \frac{36}{1350}[/tex]
[tex]Relative\ Error = 0.0267[/tex]
The percentage error is:
[tex]Percentage\ Error = Relative\ Error * 100\%[/tex]
[tex]Percentage\ Error = 0.0267* 100\%[/tex]
[tex]Percentage\ Error = 2.67\%[/tex]
