To determine the maximum possible error in density for a cubical block given the errors in the measurement of its sides and mass, we need to consider how these errors propagate through the formula for density.
1. Definition of Density:
The density [tex]\( d \)[/tex] of a material is defined as the mass [tex]\( m \)[/tex] divided by the volume [tex]\( V \)[/tex]. For a cube with side length [tex]\( l \)[/tex], the volume is [tex]\( V = l^3 \)[/tex]. Thus, the density can be written as:
[tex]\[
d = \frac{m}{l^3}
\][/tex]
2. Errors in Measurements:
- The error in the measurement of the side length [tex]\( l \)[/tex] is [tex]\( \pm 1 \% \)[/tex].
- The error in the measurement of the mass [tex]\( m \)[/tex] is [tex]\( \pm 2 \% \)[/tex].
3. Propagation of Error in Volume:
Since the volume [tex]\( V \)[/tex] of the cube is directly proportional to the cube of the side length, [tex]\( V = l^3 \)[/tex], we need to understand how the error in the side length [tex]\( l \)[/tex] affects the volume [tex]\( V \)[/tex].
- The relative error in side length [tex]\( l \)[/tex] is [tex]\( \Delta l / l = 1 \% \)[/tex].
Using the formula for relative error in volume for a cube:
[tex]\[
\frac{\Delta V}{V} = 3 \cdot \frac{\Delta l}{l}
\][/tex]
Therefore:
[tex]\[
\frac{\Delta V}{V} = 3 \times 1 \% = 3 \%
\][/tex]
4. Combining Errors for Density:
The density is mass divided by volume, so the relative error in density [tex]\( d \)[/tex] takes into account the relative errors in both mass [tex]\( m \)[/tex] and volume [tex]\( V \)[/tex].
- The relative error in mass [tex]\( m \)[/tex] is [tex]\( \Delta m / m = 2 \% \)[/tex].
- The relative error in volume [tex]\( V \)[/tex] we calculated is [tex]\( 3 \% \)[/tex].
The combined relative error in density can be calculated by summing these relative errors because they are additive for multiplication and division:
[tex]\[
\frac{\Delta d}{d} = \frac{\Delta m}{m} + \frac{\Delta V}{V}
\][/tex]
So:
[tex]\[
\frac{\Delta d}{d} = 2 \% + 3 \% = 5 \%
\][/tex]
Therefore, the maximum possible error in the density is [tex]\( \boxed{5 \%} \)[/tex].
