PART 1:
a=2% error, b=5% error, area =ab. a(1±0.02)b(1±0.05)=ab(1±0.07) approx, so the area error is about 7%.
PART 2:
If the side of the cube has length a(1±0.02) the volume is a3(1±0.02)3.
Error in volume is a3(1±0.02)3/a3=(1±0.02)3=1±0.06 approx, 6% error.
Density of cube is constant and density=mass/volume.
mass is m(1±0.03), density=m(1±0.03)/[a3(1±0.02)3].
If there were no error margin, density, d=m/a3.
Error in density=(1±0.03)/(1±0.02)3=(1±0.03)(1±0.02)-3=(1±0.03)(1∓0.06) approx.
Error=1±0.09 approx, 9% error in density.
CHECK
Suppose cube has side length 100mm with an error of 2%, then maximum length=102mm and minimum length=98mm.
Suppose mass is 100mg with an error of 3%, then maximum mass is 103mg and minimum is 97mg. Density is minimum when we have (minimum mass)/(maximum volume) and is maximum when we have (maximum mass)/(minimum volume).
Therefore 97/1023≤d≤103/983 mg/mm3, 0.0000914≤d≤0.00010944 mg/mm3. Expected density=0.0001 mg/mm3.
Therefore d ranges from 91.4% and 109.4% of its expected value. We can see 91.4% is 8.6% below expectation and 109.4 is 9.4% above.
The estimate of ±9% is reasonably accurate for the error in the density.