An engineer specifies the thermal and mechanical processing schedule for a wire-drawing operation. Given original area (A0) = 4.7000 in^2; final area (Af) = 2.8500 in^2, determine the percent cold work (CW) in %.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except CW is given, so isolate CW 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for CW:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

3. Step 3

   $$List the givens: original area (A_{0}) = 4.7000 in^2, final area (Af) = 2.8500 in^2$$

4. Step 4 — Substitute the given values:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

5. Step 5 — Evaluate:

   CW = 39.3617\ \text{%}

6. Step 6 — Check: returning CW = 39.3617 % to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 78.7234 — kept a factor of two that cancels in the correct rearrangement.
• 19.6809 — dropped that same factor in the other direction.
• 43.2979 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

A materials technician computes the percent cold work applied during thermal and mechanical processing. Given original area (A0) = 3.8500 in^2; percent cold work (CW) = 58.2000 %, determine the final area (Af) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except Af is given, so isolate Af 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for Af:

   $$Af = A_0(1-\%CW/100)$$

3. Step 3 — List the givens: original area (A0) = 3.8500 in^2, percent cold work (CW) = 58.2000 %.

4. Step 4 — Substitute the given values:

   $$Af = A_0(1-\%58.2000/100)$$

5. Step 5 — Evaluate:

   Af = 1.6093\ \text{in^2}

6. Step 6 — Check: returning Af = 1.6093 in^2 to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 3.2186 — kept a factor of two that cancels in the correct rearrangement.
• 0.8046 — dropped that same factor in the other direction.
• 1.7702 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

The thermal and mechanical processing route reduces the cross-sectional area of the bar stock. Given final area (Af) = 3.4000 in^2; percent cold work (CW) = 68.7000 %, determine the original area (A0) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except A0 is given, so isolate A0 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for A0:

   $$A_{0} = \dfrac{A_f}{1-\%CW/100}$$

3. Step 3 — List the givens: final area (Af) = 3.4000 in^2, percent cold work (CW) = 68.7000 %.

4. Step 4 — Substitute the given values:

   $$A_{0} = \dfrac{A_f}{1-\%68.7000/100}$$

5. Step 5 — Evaluate:

   A_{0} = 10.8626\ \text{in^2}

6. Step 6 — Check: returning A0 = 10.8626 in^2 to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 21.7252 — kept a factor of two that cancels in the correct rearrangement.
• 5.4313 — dropped that same factor in the other direction.
• 11.9489 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

An engineer specifies the thermal and mechanical processing schedule for a wire-drawing operation. Given original area (A0) = 4.9500 in^2; final area (Af) = 1.3500 in^2, determine the percent cold work (CW) in %.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except CW is given, so isolate CW 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for CW:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

3. Step 3

   $$List the givens: original area (A_{0}) = 4.9500 in^2, final area (Af) = 1.3500 in^2$$

4. Step 4 — Substitute the given values:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

5. Step 5 — Evaluate:

   CW = 72.7273\ \text{%}

6. Step 6 — Check: returning CW = 72.7273 % to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 145.5 — kept a factor of two that cancels in the correct rearrangement.
• 36.3636 — dropped that same factor in the other direction.
• 80.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

A materials technician computes the percent cold work applied during thermal and mechanical processing. Given original area (A0) = 1.2500 in^2; percent cold work (CW) = 86.6000 %, determine the final area (Af) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except Af is given, so isolate Af 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for Af:

   $$Af = A_0(1-\%CW/100)$$

3. Step 3 — List the givens: original area (A0) = 1.2500 in^2, percent cold work (CW) = 86.6000 %.

4. Step 4 — Substitute the given values:

   $$Af = A_0(1-\%86.6000/100)$$

5. Step 5 — Evaluate:

   Af = 0.1675\ \text{in^2}

6. Step 6 — Check: returning Af = 0.1675 in^2 to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 0.3350 — kept a factor of two that cancels in the correct rearrangement.
• 0.0838 — dropped that same factor in the other direction.
• 0.1843 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

The thermal and mechanical processing route reduces the cross-sectional area of the bar stock. Given final area (Af) = 1.8000 in^2; percent cold work (CW) = 38.8000 %, determine the original area (A0) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except A0 is given, so isolate A0 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for A0:

   $$A_{0} = \dfrac{A_f}{1-\%CW/100}$$

3. Step 3 — List the givens: final area (Af) = 1.8000 in^2, percent cold work (CW) = 38.8000 %.

4. Step 4 — Substitute the given values:

   $$A_{0} = \dfrac{A_f}{1-\%38.8000/100}$$

5. Step 5 — Evaluate:

   A_{0} = 2.9412\ \text{in^2}

6. Step 6 — Check: returning A0 = 2.9412 in^2 to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 5.8824 — kept a factor of two that cancels in the correct rearrangement.
• 1.4706 — dropped that same factor in the other direction.
• 3.2353 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

An engineer specifies the thermal and mechanical processing schedule for a wire-drawing operation. Given original area (A0) = 2.2500 in^2; final area (Af) = 3.4500 in^2, determine the percent cold work (CW) in %.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except CW is given, so isolate CW 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for CW:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

3. Step 3

   $$List the givens: original area (A_{0}) = 2.2500 in^2, final area (Af) = 3.4500 in^2$$

4. Step 4 — Substitute the given values:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

5. Step 5 — Evaluate:

   CW = -53.3333\ \text{%}

6. Step 6 — Check: returning CW = -53.3333 % to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• -106.7 — kept a factor of two that cancels in the correct rearrangement.
• -26.6667 — dropped that same factor in the other direction.
• -58.6667 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

A materials technician computes the percent cold work applied during thermal and mechanical processing. Given original area (A0) = 1.8000 in^2; percent cold work (CW) = 88.7000 %, determine the final area (Af) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except Af is given, so isolate Af 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for Af:

   $$Af = A_0(1-\%CW/100)$$

3. Step 3 — List the givens: original area (A0) = 1.8000 in^2, percent cold work (CW) = 88.7000 %.

4. Step 4 — Substitute the given values:

   $$Af = A_0(1-\%88.7000/100)$$

5. Step 5 — Evaluate:

   Af = 0.2034\ \text{in^2}

6. Step 6 — Check: returning Af = 0.2034 in^2 to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 0.4068 — kept a factor of two that cancels in the correct rearrangement.
• 0.1017 — dropped that same factor in the other direction.
• 0.2237 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

The thermal and mechanical processing route reduces the cross-sectional area of the bar stock. Given final area (Af) = 2.0000 in^2; percent cold work (CW) = 49.4000 %, determine the original area (A0) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except A0 is given, so isolate A0 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for A0:

   $$A_{0} = \dfrac{A_f}{1-\%CW/100}$$

3. Step 3 — List the givens: final area (Af) = 2.0000 in^2, percent cold work (CW) = 49.4000 %.

4. Step 4 — Substitute the given values:

   $$A_{0} = \dfrac{A_f}{1-\%49.4000/100}$$

5. Step 5 — Evaluate:

   A_{0} = 3.9526\ \text{in^2}

6. Step 6 — Check: returning A0 = 3.9526 in^2 to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 7.9051 — kept a factor of two that cancels in the correct rearrangement.
• 1.9763 — dropped that same factor in the other direction.
• 4.3478 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing

An engineer specifies the thermal and mechanical processing schedule for a wire-drawing operation. Given original area (A0) = 4.7000 in^2; final area (Af) = 0.4000 in^2, determine the percent cold work (CW) in %.

Start with the thinking

• The governing relation printed in this handbook section is Thermal and mechanical processing — cold work percentage.
• Everything except CW is given, so isolate CW 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.
• Thermal and mechanical processing of metals uses percent cold work to quantify area reduction during forming.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

2. Step 2 — Rearrange symbolically for CW:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

3. Step 3

   $$List the givens: original area (A_{0}) = 4.7000 in^2, final area (Af) = 0.4000 in^2$$

4. Step 4 — Substitute the given values:

   $$CW = \dfrac{A_0-A_f}{A_0}\times100$$

5. Step 5 — Evaluate:

   CW = 91.4894\ \text{%}

6. Step 6 — Check: returning CW = 91.4894 % to

   $$\%CW = \dfrac{A_0 - A_f}{A_0} \times 100$$

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

Why the other options are there

• 183.0 — kept a factor of two that cancels in the correct rearrangement.
• 45.7447 — dropped that same factor in the other direction.
• 100.6 — rounded an intermediate value before the final step.

Reference: FE Handbook — Thermal and Mechanical Processing