Problem 1.47 [Difficulty: 3]
Given: Soda can with estimated dimensions D = 66.0 ± 0.5 mm, H = 110 ± 0.5 mm. Soda has SG = 1.055
Find: Volume of soda in the can (based on measured mass of full and empty can); Estimate average
depth to which the can is filled and the uncertainty in the estimate.
Solution: Measurements on a can of coke give
mg,mgmmmug
fe fem
=±
=
±
∴
=
−
=
±
3865 0 50 17 5 050 369.. ..
2
1
2
2
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
=ef m
e
e
m
f
f
mu
m
m
m
m
u
m
m
m
m
u
u0.5 g
386.5 g u
mm
fe
=± =± =± =0 00129 050
175 0 0286., .
..
0019.00286.01
369
5.17
00129.01
369
5.386 2
1
22
±=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛××+
⎟
⎠
⎞
⎜
⎝
⎛××±=
m
u
Density is mass per unit volume and SG = ρ/ρΗ2Ο so
∀= = = × × × = × −
mm
HOSG gm
kg
kg
1000 g m
2
ρρ
369 1000
1
1055 350 10
363
.
The reference value ρH2O is assumed to be precise. Since SG is specified to three places beyond the decimal point,
assume uSG = ± 0.001. Then
()( )
[]
mm102
m
mm10
m
066.0
m1035044
or
4
%21.00021.0001.010019.01
3
22
36
2
2
2
1
22
2
1
22
=×
×
×=
∀
==∀
±=±=×−+×±=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
+
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
−
π
π
π
D
LL
D
u
u
SG
v
v
SG
u
m
v
v
m
u
v
SGmv
