An unsymmetric angle section requires the product of inertia for principal axis analysis. Given area (A) = 53.5000 in^2; centroidal x offset (x_bar) = -1.5000 in; centroidal y offset (y_bar) = -6.0000 in, determine the product of inertia (I_xy) in in^4.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except I_xy is given, so isolate I_xy 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for I_xy:

   $$I_{xy} = A \bar{x} \bar{y}$$

3. Step 3 — List the givens: area (A) = 53.5000 in^2, centroidal x offset (x_bar) = -1.5000 in, centroidal y offset (y_bar) = -6.0000 in.

4. Step 4 — Substitute the given values:

   $$I_{xy} = 53.5000 \bar{x} \bar{y}$$

5. Step 5 — Evaluate:

   I_{xy} = 481.5\ \text{in^4}

6. Step 6 — Check: returning I_xy = 481.5 in^4 to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

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

Reference: FE Handbook — Statics: Product of Inertia

A composite area's product of inertia about its centroid is needed before rotating axes. Given centroidal x offset (x_bar) = -5.1000 in; centroidal y offset (y_bar) = 5.0000 in; product of inertia (I_xy) = 97.0000 in^4, determine the area (A) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except A is given, so isolate A 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for A:

   $$A = \dfrac{I_{xy}}{\bar{x}\,\bar{y}}$$

3. Step 3 — List the givens: centroidal x offset (x_bar) = -5.1000 in, centroidal y offset (y_bar) = 5.0000 in, product of inertia (I_xy) = 97.0000 in^4.

4. Step 4 — Substitute the given values:

   $$A = \dfrac{97.0000}{\bar{x}\,\bar{y}}$$

5. Step 5 — Evaluate:

   A = -3.8039\ \text{in^2}

6. Step 6 — Check: returning A = -3.8039 in^2 to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

• -7.6078 — kept a factor of two that cancels in the correct rearrangement.
• -1.9020 — dropped that same factor in the other direction.
• -4.1843 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Product of Inertia

A machine part's cross section product of inertia is computed for combined bending. Given area (A) = 46.5000 in^2; centroidal y offset (y_bar) = -4.8000 in; product of inertia (I_xy) = 358.0 in^4, determine the centroidal x offset (x_bar) in in.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except x_bar is given, so isolate x_bar 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for x_bar:

   $$x_{bar} = \dfrac{I_{xy}}{A \bar{y}}$$

3. Step 3 — List the givens: area (A) = 46.5000 in^2, centroidal y offset (y_bar) = -4.8000 in, product of inertia (I_xy) = 358.0 in^4.

4. Step 4 — Substitute the given values:

   $$x_{bar} = \dfrac{358.0}{46.5000 \bar{y}}$$

5. Step 5 — Evaluate:

   $$x_{bar} = -1.6039\ \text{in}$$

6. Step 6 — Check: returning x_bar = -1.6039 in to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

• -3.2079 — kept a factor of two that cancels in the correct rearrangement.
• -0.8020 — dropped that same factor in the other direction.
• -1.7643 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Product of Inertia

An unsymmetric angle section requires the product of inertia for principal axis analysis. Given area (A) = 56.0000 in^2; centroidal x offset (x_bar) = 4.6000 in; centroidal y offset (y_bar) = -3.8000 in, determine the product of inertia (I_xy) in in^4.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except I_xy is given, so isolate I_xy 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for I_xy:

   $$I_{xy} = A \bar{x} \bar{y}$$

3. Step 3 — List the givens: area (A) = 56.0000 in^2, centroidal x offset (x_bar) = 4.6000 in, centroidal y offset (y_bar) = -3.8000 in.

4. Step 4 — Substitute the given values:

   $$I_{xy} = 56.0000 \bar{x} \bar{y}$$

5. Step 5 — Evaluate:

   I_{xy} = -978.9\ \text{in^4}

6. Step 6 — Check: returning I_xy = -978.9 in^4 to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

• -1,958 — kept a factor of two that cancels in the correct rearrangement.
• -489.4 — dropped that same factor in the other direction.
• -1,077 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Product of Inertia

A composite area's product of inertia about its centroid is needed before rotating axes. Given centroidal x offset (x_bar) = -4.8000 in; centroidal y offset (y_bar) = 0.1000 in; product of inertia (I_xy) = -278.0 in^4, determine the area (A) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except A is given, so isolate A 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for A:

   $$A = \dfrac{I_{xy}}{\bar{x}\,\bar{y}}$$

3. Step 3 — List the givens: centroidal x offset (x_bar) = -4.8000 in, centroidal y offset (y_bar) = 0.1000 in, product of inertia (I_xy) = -278.0 in^4.

4. Step 4 — Substitute the given values:

   $$A = \dfrac{-278.0}{\bar{x}\,\bar{y}}$$

5. Step 5 — Evaluate:

   A = 579.2\ \text{in^2}

6. Step 6 — Check: returning A = 579.2 in^2 to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

• 1,158 — kept a factor of two that cancels in the correct rearrangement.
• 289.6 — dropped that same factor in the other direction.
• 637.1 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Product of Inertia

A machine part's cross section product of inertia is computed for combined bending. Given area (A) = 31.5000 in^2; centroidal y offset (y_bar) = -2.0000 in; product of inertia (I_xy) = 329.0 in^4, determine the centroidal x offset (x_bar) in in.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except x_bar is given, so isolate x_bar 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for x_bar:

   $$x_{bar} = \dfrac{I_{xy}}{A \bar{y}}$$

3. Step 3 — List the givens: area (A) = 31.5000 in^2, centroidal y offset (y_bar) = -2.0000 in, product of inertia (I_xy) = 329.0 in^4.

4. Step 4 — Substitute the given values:

   $$x_{bar} = \dfrac{329.0}{31.5000 \bar{y}}$$

5. Step 5 — Evaluate:

   $$x_{bar} = -5.2222\ \text{in}$$

6. Step 6 — Check: returning x_bar = -5.2222 in to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

• -10.4444 — kept a factor of two that cancels in the correct rearrangement.
• -2.6111 — dropped that same factor in the other direction.
• -5.7444 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Product of Inertia

An unsymmetric angle section requires the product of inertia for principal axis analysis. Given area (A) = 58.5000 in^2; centroidal x offset (x_bar) = 4.8000 in; centroidal y offset (y_bar) = 0.9000 in, determine the product of inertia (I_xy) in in^4.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except I_xy is given, so isolate I_xy 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for I_xy:

   $$I_{xy} = A \bar{x} \bar{y}$$

3. Step 3 — List the givens: area (A) = 58.5000 in^2, centroidal x offset (x_bar) = 4.8000 in, centroidal y offset (y_bar) = 0.9000 in.

4. Step 4 — Substitute the given values:

   $$I_{xy} = 58.5000 \bar{x} \bar{y}$$

5. Step 5 — Evaluate:

   I_{xy} = 252.7\ \text{in^4}

6. Step 6 — Check: returning I_xy = 252.7 in^4 to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

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

Reference: FE Handbook — Statics: Product of Inertia

A composite area's product of inertia about its centroid is needed before rotating axes. Given centroidal x offset (x_bar) = 1.8000 in; centroidal y offset (y_bar) = 3.0000 in; product of inertia (I_xy) = -182.0 in^4, determine the area (A) in in^2.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except A is given, so isolate A 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for A:

   $$A = \dfrac{I_{xy}}{\bar{x}\,\bar{y}}$$

3. Step 3 — List the givens: centroidal x offset (x_bar) = 1.8000 in, centroidal y offset (y_bar) = 3.0000 in, product of inertia (I_xy) = -182.0 in^4.

4. Step 4 — Substitute the given values:

   $$A = \dfrac{-182.0}{\bar{x}\,\bar{y}}$$

5. Step 5 — Evaluate:

   A = -33.7037\ \text{in^2}

6. Step 6 — Check: returning A = -33.7037 in^2 to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

• -67.4074 — kept a factor of two that cancels in the correct rearrangement.
• -16.8519 — dropped that same factor in the other direction.
• -37.0741 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Product of Inertia

A machine part's cross section product of inertia is computed for combined bending. Given area (A) = 58.0000 in^2; centroidal y offset (y_bar) = -2.8000 in; product of inertia (I_xy) = 154.0 in^4, determine the centroidal x offset (x_bar) in in.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except x_bar is given, so isolate x_bar 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for x_bar:

   $$x_{bar} = \dfrac{I_{xy}}{A \bar{y}}$$

3. Step 3 — List the givens: area (A) = 58.0000 in^2, centroidal y offset (y_bar) = -2.8000 in, product of inertia (I_xy) = 154.0 in^4.

4. Step 4 — Substitute the given values:

   $$x_{bar} = \dfrac{154.0}{58.0000 \bar{y}}$$

5. Step 5 — Evaluate:

   $$x_{bar} = -0.9483\ \text{in}$$

6. Step 6 — Check: returning x_bar = -0.9483 in to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

• -1.8966 — kept a factor of two that cancels in the correct rearrangement.
• -0.4741 — dropped that same factor in the other direction.
• -1.0431 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Product of Inertia

An unsymmetric angle section requires the product of inertia for principal axis analysis. Given area (A) = 31.5000 in^2; centroidal x offset (x_bar) = 3.6000 in; centroidal y offset (y_bar) = 3.7000 in, determine the product of inertia (I_xy) in in^4.

Start with the thinking

• The governing relation printed in this handbook section is Product of inertia (rectangle).
• Everything except I_xy is given, so isolate I_xy 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 product of inertia of an area about centroidal axes is required for unsymmetric bending and rotated-axis transformations.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$I_{xy} = A \bar{x} \bar{y}$$

2. Step 2 — Rearrange symbolically for I_xy:

   $$I_{xy} = A \bar{x} \bar{y}$$

3. Step 3 — List the givens: area (A) = 31.5000 in^2, centroidal x offset (x_bar) = 3.6000 in, centroidal y offset (y_bar) = 3.7000 in.

4. Step 4 — Substitute the given values:

   $$I_{xy} = 31.5000 \bar{x} \bar{y}$$

5. Step 5 — Evaluate:

   I_{xy} = 419.6\ \text{in^4}

6. Step 6 — Check: returning I_xy = 419.6 in^4 to

   $$I_{xy} = A \bar{x} \bar{y}$$

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

Why the other options are there

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

Reference: FE Handbook — Statics: Product of Inertia