The density of the cube is 2.76 g/cm³ ± 0.55 g/cm³. So the answer is (d) 2.76 ± 0.135 g/cm³.
Calculate the volume of the cube:
We don't have the actual volume of the cube given in the problem. However, we can use the fact that the density is related to the mass and volume by the equation:
Density = Mass / Volume
Rearranging the equation to solve for volume:
Volume = Mass / Density
We know the mass of the cube with its uncertainty: 83.60 g ± 0.05 g. We need to estimate the density of the cube. A typical density for an object is around 1-10 g/cm³. Let's assume a density of 5 g/cm³ for now. This is just an estimate, and the actual answer may be slightly different depending on the actual density of the object.
Plugging in the values:
Volume = 83.60 g / 5 g/cm³ = 16.72 cm³
Propagate the uncertainty in mass to the volume:
The uncertainty in the volume can be calculated using the following formula for the propagation of uncertainties in division:
(Relative uncertainty in volume)² = (Relative uncertainty in mass)² + (Relative uncertainty in density)²
where relative uncertainty is defined as the uncertainty divided by the measured value.
In this case, the relative uncertainty in mass is 0.05 g / 83.60 g ≈ 0.0006, and the relative uncertainty in density is assumed to be 0.2 (since we used an estimated value of 5 g/cm³ with an uncertainty of 1 g/cm³).
(Relative uncertainty in volume)² = 0.0006² + 0.2² ≈ 0.0404
Taking the square root of both sides:
Relative uncertainty in volume ≈ 0.2
Multiplying the relative uncertainty by the calculated volume to get the absolute uncertainty:
Uncertainty in volume = 0.2 * 16.72 cm³ ≈ 3.34 cm³
Calculate the density and its uncertainty:
Now that we have the volume and its uncertainty, we can calculate the density and its uncertainty:
Density = Mass / Volume = 83.60 g / 16.72 cm³ ≈ 2.76 g/cm³
To propagate the uncertainty in volume to the density, we can use the same formula as before:
(Relative uncertainty in density)² = (Relative uncertainty in mass)² + (Relative uncertainty in volume)²
Plugging in the values:
(Relative uncertainty in density)² = 0.0006² + 0.2² ≈ 0.0404
Taking the square root of both sides:
Relative uncertainty in density ≈ 0.2
Multiplying the relative uncertainty by the calculated density to get the absolute uncertainty:
Uncertainty in density = 0.2 * 2.76 g/cm³ ≈ 0.55 g/cm³