Problem 1.49 [Difficulty: 4]
Given: Dimensions of soda can: D = 66 mm, H = 110 mm
Find: Measurement precision needed to allow volume to be estimated with an
uncertainty of ± 0.5 percent or less.
Solution: Use the methods of Appendix F:
Computing equations:
1
2
2
22
HD
DH
4
HD
uuu
HD
π
∀
∀=
⎡⎤
∂∀ ∂∀
⎛⎞⎛⎞
=± +
⎢⎥
⎜⎟⎜⎟
∀∂ ∀∂
⎝⎠⎝⎠
⎢⎥
⎣⎦
Since 2
DH
4
π
∀= , then 2
D
H4
π
∂∀
∂=and DH
D2
π
∂∀
∂=. Letting DD
ux
δ
=
±and HH
ux
δ
=
±, and substituting,
11
22
2222
2
22
4H D 4D DH 2
uDH 4 H DH 2 D H D
xxxx
πδ π δ δ δ
ππ
∀
⎡⎤
⎡⎤
⎛⎞
⎛⎞⎛⎞⎛⎞
⎢⎥
=± + =± +
⎢⎥
⎜⎟
⎜⎟⎜⎟⎜⎟
⎝⎠⎝⎠⎝⎠
⎢⎥
⎢⎥
⎝⎠ ⎣⎦
⎣⎦
Solving,
22 22
22
212
u()
HD HD
xx
x
δδ
δ
∀
⎡
⎤
⎛⎞⎛ ⎞ ⎛⎞⎛⎞
=+ = +
⎢
⎥
⎜⎟⎜ ⎟ ⎜⎟⎜⎟
⎝⎠⎝ ⎠ ⎝⎠⎝⎠
⎢
⎥
⎣
⎦
()()
11
22
22 2 2
12 12
HD 110mm 66mm
u0.005 0.158 mm
() ()
x
δ
∀
=± =± =±
⎡⎤⎡ ⎤
++
⎣⎦
⎢⎥
⎣⎦
Check:
3
H
3
D
0.158 mm
u1.4410
H 110 mm
0.158 mm
u2.3910
D66mm
x
x
δ
δ
−
−
=± =± =± ×
=± =± =± ×
11
22
22 2 2
HD
u [(u ) (2u ) ] [(0.00144) (0.00478) ] 0.00499
∀=± + =± + =±
If δx represents half the least count, a minimum resolution of about 2 δx ≈ 0.32 mm is needed.
H
D