A calibration lab combines two sources of measurement uncertainty. Given uncertainty of quantity 1 (u1) = 2.4000; uncertainty of quantity 2 (u2) = 0.7500, determine the combined uncertainty (uc).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except uc is given, so isolate uc symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for uc:

   $$uc = \sqrt{u_1^2+u_2^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 1 (u_{1}) = 2.4000, uncertainty of quantity 2 (u_{2}) = 0.7500$$

4. Step 4 — Substitute the given values:

   $$uc = \sqrt{u_1^2+u_2^2}$$

5. Step 5 — Evaluate:

   $$uc = 2.5145$$

6. Step 6 — Check: returning uc = 2.5145 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 5.0289 — kept a factor of two that cancels in the correct rearrangement.
• 1.2572 — dropped that same factor in the other direction.
• 2.7659 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

An engineer reports the measurement uncertainty of an instrument reading. Given uncertainty of quantity 2 (u2) = 0.8500; combined uncertainty (uc) = 7.0500, determine the uncertainty of quantity 1 (u1).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except u1 is given, so isolate u1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for u1:

   $$u_{1} = \sqrt{u_c^2-u_2^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 2 (u_{2}) = 0.8500, combined uncertainty (uc) = 7.0500$$

4. Step 4 — Substitute the given values:

   $$u_{1} = \sqrt{u_c^2-u_2^2}$$

5. Step 5 — Evaluate:

   $$u_{1} = 6.9986$$

6. Step 6 — Check: returning u1 = 6.9986 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 13.9971 — kept a factor of two that cancels in the correct rearrangement.
• 3.4993 — dropped that same factor in the other direction.
• 7.6984 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

The measurement uncertainty of a length gauge is propagated from two components. Given uncertainty of quantity 1 (u1) = 1.8500; combined uncertainty (uc) = 2.0900, determine the uncertainty of quantity 2 (u2).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except u2 is given, so isolate u2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for u2:

   $$u_{2} = \sqrt{u_c^2-u_1^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 1 (u_{1}) = 1.8500, combined uncertainty (uc) = 2.0900$$

4. Step 4 — Substitute the given values:

   $$u_{2} = \sqrt{u_c^2-u_1^2}$$

5. Step 5 — Evaluate:

   $$u_{2} = 0.9724$$

6. Step 6 — Check: returning u2 = 0.9724 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 1.9448 — kept a factor of two that cancels in the correct rearrangement.
• 0.4862 — dropped that same factor in the other direction.
• 1.0697 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

A calibration lab combines two sources of measurement uncertainty. Given uncertainty of quantity 1 (u1) = 4.3000; uncertainty of quantity 2 (u2) = 2.6000, determine the combined uncertainty (uc).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except uc is given, so isolate uc symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for uc:

   $$uc = \sqrt{u_1^2+u_2^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 1 (u_{1}) = 4.3000, uncertainty of quantity 2 (u_{2}) = 2.6000$$

4. Step 4 — Substitute the given values:

   $$uc = \sqrt{u_1^2+u_2^2}$$

5. Step 5 — Evaluate:

   $$uc = 5.0249$$

6. Step 6 — Check: returning uc = 5.0249 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 10.0499 — kept a factor of two that cancels in the correct rearrangement.
• 2.5125 — dropped that same factor in the other direction.
• 5.5274 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

An engineer reports the measurement uncertainty of an instrument reading. Given uncertainty of quantity 2 (u2) = 4.2000; combined uncertainty (uc) = 2.6800, determine the uncertainty of quantity 1 (u1).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except u1 is given, so isolate u1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for u1:

   $$u_{1} = \sqrt{u_c^2-u_2^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 2 (u_{2}) = 4.2000, combined uncertainty (uc) = 2.6800$$

4. Step 4 — Substitute the given values:

   $$u_{1} = \sqrt{u_c^2-u_2^2}$$

5. Step 5 — Evaluate:

   $$u_{1} = 0.0100$$

6. Step 6 — Check: returning u1 = 0.0100 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0200 — kept a factor of two that cancels in the correct rearrangement.
• 0.0050 — dropped that same factor in the other direction.
• 0.0110 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

The measurement uncertainty of a length gauge is propagated from two components. Given uncertainty of quantity 1 (u1) = 0.6500; combined uncertainty (uc) = 2.7600, determine the uncertainty of quantity 2 (u2).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except u2 is given, so isolate u2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for u2:

   $$u_{2} = \sqrt{u_c^2-u_1^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 1 (u_{1}) = 0.6500, combined uncertainty (uc) = 2.7600$$

4. Step 4 — Substitute the given values:

   $$u_{2} = \sqrt{u_c^2-u_1^2}$$

5. Step 5 — Evaluate:

   $$u_{2} = 2.6824$$

6. Step 6 — Check: returning u2 = 2.6824 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 5.3647 — kept a factor of two that cancels in the correct rearrangement.
• 1.3412 — dropped that same factor in the other direction.
• 2.9506 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

A calibration lab combines two sources of measurement uncertainty. Given uncertainty of quantity 1 (u1) = 1.4500; uncertainty of quantity 2 (u2) = 1.3000, determine the combined uncertainty (uc).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except uc is given, so isolate uc symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for uc:

   $$uc = \sqrt{u_1^2+u_2^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 1 (u_{1}) = 1.4500, uncertainty of quantity 2 (u_{2}) = 1.3000$$

4. Step 4 — Substitute the given values:

   $$uc = \sqrt{u_1^2+u_2^2}$$

5. Step 5 — Evaluate:

   $$uc = 1.9474$$

6. Step 6 — Check: returning uc = 1.9474 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 3.8949 — kept a factor of two that cancels in the correct rearrangement.
• 0.9737 — dropped that same factor in the other direction.
• 2.1422 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

An engineer reports the measurement uncertainty of an instrument reading. Given uncertainty of quantity 2 (u2) = 1.6500; combined uncertainty (uc) = 2.2000, determine the uncertainty of quantity 1 (u1).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except u1 is given, so isolate u1 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for u1:

   $$u_{1} = \sqrt{u_c^2-u_2^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 2 (u_{2}) = 1.6500, combined uncertainty (uc) = 2.2000$$

4. Step 4 — Substitute the given values:

   $$u_{1} = \sqrt{u_c^2-u_2^2}$$

5. Step 5 — Evaluate:

   $$u_{1} = 1.4552$$

6. Step 6 — Check: returning u1 = 1.4552 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 2.9103 — kept a factor of two that cancels in the correct rearrangement.
• 0.7276 — dropped that same factor in the other direction.
• 1.6007 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

The measurement uncertainty of a length gauge is propagated from two components. Given uncertainty of quantity 1 (u1) = 3.7000; combined uncertainty (uc) = 2.2300, determine the uncertainty of quantity 2 (u2).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except u2 is given, so isolate u2 symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for u2:

   $$u_{2} = \sqrt{u_c^2-u_1^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 1 (u_{1}) = 3.7000, combined uncertainty (uc) = 2.2300$$

4. Step 4 — Substitute the given values:

   $$u_{2} = \sqrt{u_c^2-u_1^2}$$

5. Step 5 — Evaluate:

   $$u_{2} = 0.0100$$

6. Step 6 — Check: returning u2 = 0.0100 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.0200 — kept a factor of two that cancels in the correct rearrangement.
• 0.0050 — dropped that same factor in the other direction.
• 0.0110 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty

A calibration lab combines two sources of measurement uncertainty. Given uncertainty of quantity 1 (u1) = 2.9500; uncertainty of quantity 2 (u2) = 4.6500, determine the combined uncertainty (uc).

Start with the thinking

• The governing relation printed in this handbook section is Measurement uncertainty propagation.
• Everything except uc is given, so isolate uc symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• Measurement uncertainty from independent sources combines in quadrature to give the combined uncertainty.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

2. Step 2 — Rearrange symbolically for uc:

   $$uc = \sqrt{u_1^2+u_2^2}$$

3. Step 3

   $$List the givens: uncertainty of quantity 1 (u_{1}) = 2.9500, uncertainty of quantity 2 (u_{2}) = 4.6500$$

4. Step 4 — Substitute the given values:

   $$uc = \sqrt{u_1^2+u_2^2}$$

5. Step 5 — Evaluate:

   $$uc = 5.5068$$

6. Step 6 — Check: returning uc = 5.5068 to

   $$u_c = \sqrt{u_1^2 + u_2^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 11.0136 — kept a factor of two that cancels in the correct rearrangement.
• 2.7534 — dropped that same factor in the other direction.
• 6.0575 — rounded an intermediate value before the final step.

Reference: FE Handbook — Measurement Uncertainty