Certainly! Here is the step-by-step solution for calculating the error in the measurement of the density of a cube given the errors in the measurements of mass and side length.
### Step-by-Step Solution:
1. Understand the Formula for Density:
The density ([tex]\(\rho\)[/tex]) of a cube is given by the formula:
[tex]\[
\rho = \frac{m}{V}
\][/tex]
where [tex]\(m\)[/tex] is the mass and [tex]\(V\)[/tex] is the volume of the cube. The volume of a cube is [tex]\(V = l^3\)[/tex], where [tex]\(l\)[/tex] is the length of the side of the cube. Therefore:
[tex]\[
\rho = \frac{m}{l^3}
\][/tex]
2. Determine Relevant Errors:
We are given:
- The maximum error in the measurement of mass ([tex]\(m\)[/tex]) is [tex]\(3 \%\)[/tex].
- The maximum error in the measurement of length ([tex]\(l\)[/tex]) is [tex]\(2 \%\)[/tex].
3. Relative Error in Density:
When combining errors in different measurements, the relative error in density can be approximated using the propagation of errors formula for division and powers:
[tex]\[
\frac{\delta \rho}{\rho} \approx \frac{\delta m}{m} + 3 \times \frac{\delta l}{l}
\][/tex]
Here:
- [tex]\(\frac{\delta \rho}{\rho}\)[/tex] represents the relative error in density.
- [tex]\(\frac{\delta m}{m}\)[/tex] represents the relative error in mass ([tex]\(3\%\)[/tex]).
- [tex]\(\frac{\delta l}{l}\)[/tex] represents the relative error in length ([tex]\(2\%\)[/tex]).
- The factor of 3 comes from the power to which [tex]\(l\)[/tex] is raised in the volume equation ([tex]\(l^3\)[/tex]).
4. Substitute Given Errors:
Substituting the given errors:
[tex]\[
\frac{\delta \rho}{\rho} \approx \frac{3 \%}{100} + 3 \times \frac{2 \%}{100}
\][/tex]
5. Calculate Combined Error:
Simplify the expression:
[tex]\[
\frac{\delta \rho}{\rho} \approx 3 \% + 3 \times 2 \%
\][/tex]
[tex]\[
\frac{\delta \rho}{\rho} \approx 3 \% + 6 \%
\][/tex]
[tex]\[
\frac{\delta \rho}{\rho} \approx 9 \%
\][/tex]
6. Result:
Therefore, the maximum error in the measurement of the density of the cube is:
[tex]\[
9 \%
\][/tex]
Hence, the maximum error in the measurement of the density of the cube is [tex]\(9\%\)[/tex].
