Significant figures?

Sig figs ....

The density written as 2.937 g/cm³ is not incorrect. It has one too-many significant digits.

Mass of metal = 45.81g (4 significant digits)

Volume of metal = 119.3 - 103.7 = 15.6 cm³ (3 significant digits ... when doing addition or subtraction we go by the rightmost least significant digit. That is covered in any good set of significant digit rules.)

Density = m / V = 45.81g / 15.6 cm³ = 2.94 g/cm³ (express to 3 significant digits)

(Volumes in milliliters are usually reserved for fluids. Cubic centimeters work for all substances, and normally we would use cubic centimeters for solids.)

If you are going to go by the significant digit rules, then the answer is given as 2.94 g/cm³. The use of the significant digit rules does not require the propagation of uncertainty and and an uncertainty for the answer.

Expressing the the answer along with "+/- some percentage" is not the norm.

Propagation of errors requires knowing the uncertainty in each measurement. Short of that we can assume the uncertainty is +/- 0.01g for the mass and +/- 0.1 mL for the volume.

uncertainty in mass = +/- 0.01 ..................... ..................... 45.81 +/- 0.01

uncertainty in volume = √(0.1^2 + 0.1^2) = +/- 0.14 .......... 15.6 +/- 0.14

(Uncertainty in addition and subtraction is the square root of the sum of the squares of the uncertainties. Usually the uncertainty is expressed with one significant digit, except when the first digit of the uncertainty is "1".)

Density = m / V = (45.81 +/- 0.01) / (15.6 +/- 0.14) = 2.94 +/- ???

The uncertainty in division is...

δR = R(√(δX/X)² + (δY/Y)²)

δR = 2.94(√(δ.01/45.81)² + (0.14/15.6)²) = 0.03 ............ round to 1 significant digit

Density = 2.94 +/- 0.03 g/cm³ .... therefore, the density is rounded to the hundredth's column since the "3" in 0.03 is in the hundredths column. A density of 2.937 g/cm³ would be incorrect. The sig fig rules usually do a pretty good job of getting the number of significant digits right in the answer.

2.94 +/- 0.03 g/cm³.... this is essentially the same as saying 2.94 +/- 1%

But as you can see, the propagation of uncertainty can be involved and time-consuming and usually gives the same answer as following the sig fig rules.

The printing is very hard to read, but looks like your answer is correct.

or the same answer using SN

Their answer is not wrong, just not what most of us would answer..