A metallurgist determines the ASTM grain size number from a micrograph count. Given ASTM grain size number (n) = 10.0000, determine the grains per sq. inch at 100x (N).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except N is given, so isolate N 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for N:

   $$N = 2^{n-1}$$

3. Step 3

   $$List the givens: ASTM grain size number (n) = 10.0000$$

4. Step 4 — Substitute the given values:

   $$N = 2^{10.0000-1}$$

5. Step 5 — Evaluate:

   $$N = 512.0$$

6. Step 6 — Check: returning N = 512.0 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 1,024 — kept a factor of two that cancels in the correct rearrangement.
• 256.0 — dropped that same factor in the other direction.
• 563.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

An engineer specifies an ASTM grain size number for a forged component. Given grains per sq. inch at 100x (N) = 232.0, determine the ASTM grain size number (n).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except n is given, so isolate n 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \log_2(N) + 1$$

3. Step 3

   $$List the givens: grains per sq. inch at 100x (N) = 232.0$$

4. Step 4 — Substitute the given values:

   $$n = \log_2(232.0) + 1$$

5. Step 5 — Evaluate:

   $$n = 8.8580$$

6. Step 6 — Check: returning n = 8.8580 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 17.7160 — kept a factor of two that cancels in the correct rearrangement.
• 4.4290 — dropped that same factor in the other direction.
• 9.7438 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

The ASTM grain size number is used to compare heat treatments of two alloys. Given ASTM grain size number (n) = 2.0000, determine the grains per sq. inch at 100x (N).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except N is given, so isolate N 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for N:

   $$N = 2^{n-1}$$

3. Step 3

   $$List the givens: ASTM grain size number (n) = 2.0000$$

4. Step 4 — Substitute the given values:

   $$N = 2^{2.0000-1}$$

5. Step 5 — Evaluate:

   $$N = 2.0000$$

6. Step 6 — Check: returning N = 2.0000 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 4.0000 — kept a factor of two that cancels in the correct rearrangement.
• 1.0000 — dropped that same factor in the other direction.
• 2.2000 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

A metallurgist determines the ASTM grain size number from a micrograph count. Given grains per sq. inch at 100x (N) = 20.0000, determine the ASTM grain size number (n).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except n is given, so isolate n 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \log_2(N) + 1$$

3. Step 3

   $$List the givens: grains per sq. inch at 100x (N) = 20.0000$$

4. Step 4 — Substitute the given values:

   $$n = \log_2(20.0000) + 1$$

5. Step 5 — Evaluate:

   $$n = 5.3219$$

6. Step 6 — Check: returning n = 5.3219 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 10.6439 — kept a factor of two that cancels in the correct rearrangement.
• 2.6610 — dropped that same factor in the other direction.
• 5.8541 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

An engineer specifies an ASTM grain size number for a forged component. Given ASTM grain size number (n) = 9.5000, determine the grains per sq. inch at 100x (N).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except N is given, so isolate N 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for N:

   $$N = 2^{n-1}$$

3. Step 3

   $$List the givens: ASTM grain size number (n) = 9.5000$$

4. Step 4 — Substitute the given values:

   $$N = 2^{9.5000-1}$$

5. Step 5 — Evaluate:

   $$N = 362.0$$

6. Step 6 — Check: returning N = 362.0 to

   $$N = 2^{n-1}$$

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

Why the other options are there

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

Reference: FE Handbook — ASTM Grain Size

The ASTM grain size number is used to compare heat treatments of two alloys. Given grains per sq. inch at 100x (N) = 58.0000, determine the ASTM grain size number (n).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except n is given, so isolate n 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \log_2(N) + 1$$

3. Step 3

   $$List the givens: grains per sq. inch at 100x (N) = 58.0000$$

4. Step 4 — Substitute the given values:

   $$n = \log_2(58.0000) + 1$$

5. Step 5 — Evaluate:

   $$n = 6.8580$$

6. Step 6 — Check: returning n = 6.8580 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 13.7160 — kept a factor of two that cancels in the correct rearrangement.
• 3.4290 — dropped that same factor in the other direction.
• 7.5438 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

A metallurgist determines the ASTM grain size number from a micrograph count. Given ASTM grain size number (n) = 5.0000, determine the grains per sq. inch at 100x (N).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except N is given, so isolate N 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for N:

   $$N = 2^{n-1}$$

3. Step 3

   $$List the givens: ASTM grain size number (n) = 5.0000$$

4. Step 4 — Substitute the given values:

   $$N = 2^{5.0000-1}$$

5. Step 5 — Evaluate:

   $$N = 16.0000$$

6. Step 6 — Check: returning N = 16.0000 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 32.0000 — kept a factor of two that cancels in the correct rearrangement.
• 8.0000 — dropped that same factor in the other direction.
• 17.6000 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

An engineer specifies an ASTM grain size number for a forged component. Given grains per sq. inch at 100x (N) = 68.0000, determine the ASTM grain size number (n).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except n is given, so isolate n 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \log_2(N) + 1$$

3. Step 3

   $$List the givens: grains per sq. inch at 100x (N) = 68.0000$$

4. Step 4 — Substitute the given values:

   $$n = \log_2(68.0000) + 1$$

5. Step 5 — Evaluate:

   $$n = 7.0875$$

6. Step 6 — Check: returning n = 7.0875 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 14.1749 — kept a factor of two that cancels in the correct rearrangement.
• 3.5437 — dropped that same factor in the other direction.
• 7.7962 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

The ASTM grain size number is used to compare heat treatments of two alloys. Given ASTM grain size number (n) = 4.5000, determine the grains per sq. inch at 100x (N).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except N is given, so isolate N 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for N:

   $$N = 2^{n-1}$$

3. Step 3

   $$List the givens: ASTM grain size number (n) = 4.5000$$

4. Step 4 — Substitute the given values:

   $$N = 2^{4.5000-1}$$

5. Step 5 — Evaluate:

   $$N = 11.3137$$

6. Step 6 — Check: returning N = 11.3137 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 22.6274 — kept a factor of two that cancels in the correct rearrangement.
• 5.6569 — dropped that same factor in the other direction.
• 12.4451 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size

A metallurgist determines the ASTM grain size number from a micrograph count. Given grains per sq. inch at 100x (N) = 13.0000, determine the ASTM grain size number (n).

Start with the thinking

• The governing relation printed in this handbook section is ASTM grain size number.
• Everything except n is given, so isolate n 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.
• The ASTM grain size number relates a material's grain size index to the number of grains observed per unit area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$N = 2^{n-1}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \log_2(N) + 1$$

3. Step 3

   $$List the givens: grains per sq. inch at 100x (N) = 13.0000$$

4. Step 4 — Substitute the given values:

   $$n = \log_2(13.0000) + 1$$

5. Step 5 — Evaluate:

   $$n = 4.7004$$

6. Step 6 — Check: returning n = 4.7004 to

   $$N = 2^{n-1}$$

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

Why the other options are there

• 9.4009 — kept a factor of two that cancels in the correct rearrangement.
• 2.3502 — dropped that same factor in the other direction.
• 5.1705 — rounded an intermediate value before the final step.

Reference: FE Handbook — ASTM Grain Size