An angle bracket made of two rectangular plates needs its centroid located. Given area 1 (A_1) = 19.0000 in^2; centroid 1 x (x_1) = 3.5000 in; area 2 (A_2) = 36.5000 in^2; centroid 2 x (x_2) = 14.2000 in, determine the composite centroid x (x_bar) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_bar:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

3. Step 3 — List the givens: area 1 (A_1) = 19.0000 in^2, centroid 1 x (x_1) = 3.5000 in, area 2 (A_2) = 36.5000 in^2, centroid 2 x (x_2) = 14.2000 in.

4. Step 4 — Substitute the given values:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

5. Step 5 — Evaluate:

   $$x_{bar} = 10.5369\ \text{in}$$

6. Step 6 — Check: returning x_bar = 10.5369 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 21.0739 — kept a factor of two that cancels in the correct rearrangement.
• 5.2685 — dropped that same factor in the other direction.
• 11.5906 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

A composite bridge deck cross section is split into rectangles to find the overall centroid. Given area 1 (A_1) = 24.0000 in^2; area 2 (A_2) = 22.0000 in^2; centroid 2 x (x_2) = 5.3000 in; composite centroid x (x_bar) = 8.5800 in, determine the centroid 1 x (x_1) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• Everything except x_1 is given, so isolate x_1 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_1:

   $$x_{1} = \dfrac{\bar{x}(A_1+A_2) - A_2 x_2}{A_1}$$

3. Step 3 — List the givens: area 1 (A_1) = 24.0000 in^2, area 2 (A_2) = 22.0000 in^2, centroid 2 x (x_2) = 5.3000 in, composite centroid x (x_bar) = 8.5800 in.

4. Step 4 — Substitute the given values:

   $$x_{1} = \dfrac{\bar{x}(A_1+A_2) - A_2 x_2}{A_1}$$

5. Step 5 — Evaluate:

   $$x_{1} = 11.5867\ \text{in}$$

6. Step 6 — Check: returning x_1 = 11.5867 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 23.1733 — kept a factor of two that cancels in the correct rearrangement.
• 5.7933 — dropped that same factor in the other direction.
• 12.7453 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

A machined bracket's area is decomposed into simple shapes to compute the centroid. Given area 1 (A_1) = 36.0000 in^2; centroid 1 x (x_1) = 6.7000 in; area 2 (A_2) = 7.5000 in^2; composite centroid x (x_bar) = 2.1900 in, determine the centroid 2 x (x_2) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• Everything except x_2 is given, so isolate x_2 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_2:

   $$x_{2} = \dfrac{\bar{x}(A_1+A_2) - A_1 x_1}{A_2}$$

3. Step 3 — List the givens: area 1 (A_1) = 36.0000 in^2, centroid 1 x (x_1) = 6.7000 in, area 2 (A_2) = 7.5000 in^2, composite centroid x (x_bar) = 2.1900 in.

4. Step 4 — Substitute the given values:

   $$x_{2} = \dfrac{\bar{x}(A_1+A_2) - A_1 x_1}{A_2}$$

5. Step 5 — Evaluate:

   $$x_{2} = -19.4580\ \text{in}$$

6. Step 6 — Check: returning x_2 = -19.4580 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• -38.9160 — kept a factor of two that cancels in the correct rearrangement.
• -9.7290 — dropped that same factor in the other direction.
• -21.4038 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

An angle bracket made of two rectangular plates needs its centroid located. Given area 1 (A_1) = 11.0000 in^2; centroid 1 x (x_1) = 7.0000 in; area 2 (A_2) = 26.5000 in^2; centroid 2 x (x_2) = 3.3000 in, determine the composite centroid x (x_bar) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_bar:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

3. Step 3 — List the givens: area 1 (A_1) = 11.0000 in^2, centroid 1 x (x_1) = 7.0000 in, area 2 (A_2) = 26.5000 in^2, centroid 2 x (x_2) = 3.3000 in.

4. Step 4 — Substitute the given values:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

5. Step 5 — Evaluate:

   $$x_{bar} = 4.3853\ \text{in}$$

6. Step 6 — Check: returning x_bar = 4.3853 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 8.7707 — kept a factor of two that cancels in the correct rearrangement.
• 2.1927 — dropped that same factor in the other direction.
• 4.8239 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

A composite bridge deck cross section is split into rectangles to find the overall centroid. Given area 1 (A_1) = 12.0000 in^2; area 2 (A_2) = 16.5000 in^2; centroid 2 x (x_2) = 9.0000 in; composite centroid x (x_bar) = 6.1100 in, determine the centroid 1 x (x_1) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• Everything except x_1 is given, so isolate x_1 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_1:

   $$x_{1} = \dfrac{\bar{x}(A_1+A_2) - A_2 x_2}{A_1}$$

3. Step 3 — List the givens: area 1 (A_1) = 12.0000 in^2, area 2 (A_2) = 16.5000 in^2, centroid 2 x (x_2) = 9.0000 in, composite centroid x (x_bar) = 6.1100 in.

4. Step 4 — Substitute the given values:

   $$x_{1} = \dfrac{\bar{x}(A_1+A_2) - A_2 x_2}{A_1}$$

5. Step 5 — Evaluate:

   $$x_{1} = 2.1363\ \text{in}$$

6. Step 6 — Check: returning x_1 = 2.1363 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 4.2725 — kept a factor of two that cancels in the correct rearrangement.
• 1.0681 — dropped that same factor in the other direction.
• 2.3499 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

A machined bracket's area is decomposed into simple shapes to compute the centroid. Given area 1 (A_1) = 5.5000 in^2; centroid 1 x (x_1) = 4.1000 in; area 2 (A_2) = 14.0000 in^2; composite centroid x (x_bar) = 12.8100 in, determine the centroid 2 x (x_2) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• Everything except x_2 is given, so isolate x_2 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_2:

   $$x_{2} = \dfrac{\bar{x}(A_1+A_2) - A_1 x_1}{A_2}$$

3. Step 3 — List the givens: area 1 (A_1) = 5.5000 in^2, centroid 1 x (x_1) = 4.1000 in, area 2 (A_2) = 14.0000 in^2, composite centroid x (x_bar) = 12.8100 in.

4. Step 4 — Substitute the given values:

   $$x_{2} = \dfrac{\bar{x}(A_1+A_2) - A_1 x_1}{A_2}$$

5. Step 5 — Evaluate:

   $$x_{2} = 16.2318\ \text{in}$$

6. Step 6 — Check: returning x_2 = 16.2318 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 32.4636 — kept a factor of two that cancels in the correct rearrangement.
• 8.1159 — dropped that same factor in the other direction.
• 17.8550 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

An angle bracket made of two rectangular plates needs its centroid located. Given area 1 (A_1) = 31.5000 in^2; centroid 1 x (x_1) = 1.3000 in; area 2 (A_2) = 24.5000 in^2; centroid 2 x (x_2) = 9.8000 in, determine the composite centroid x (x_bar) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_bar:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

3. Step 3 — List the givens: area 1 (A_1) = 31.5000 in^2, centroid 1 x (x_1) = 1.3000 in, area 2 (A_2) = 24.5000 in^2, centroid 2 x (x_2) = 9.8000 in.

4. Step 4 — Substitute the given values:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

5. Step 5 — Evaluate:

   $$x_{bar} = 5.0188\ \text{in}$$

6. Step 6 — Check: returning x_bar = 5.0188 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 10.0375 — kept a factor of two that cancels in the correct rearrangement.
• 2.5094 — dropped that same factor in the other direction.
• 5.5206 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

A composite bridge deck cross section is split into rectangles to find the overall centroid. Given area 1 (A_1) = 8.0000 in^2; area 2 (A_2) = 13.5000 in^2; centroid 2 x (x_2) = 5.0000 in; composite centroid x (x_bar) = 4.5200 in, determine the centroid 1 x (x_1) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• Everything except x_1 is given, so isolate x_1 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_1:

   $$x_{1} = \dfrac{\bar{x}(A_1+A_2) - A_2 x_2}{A_1}$$

3. Step 3 — List the givens: area 1 (A_1) = 8.0000 in^2, area 2 (A_2) = 13.5000 in^2, centroid 2 x (x_2) = 5.0000 in, composite centroid x (x_bar) = 4.5200 in.

4. Step 4 — Substitute the given values:

   $$x_{1} = \dfrac{\bar{x}(A_1+A_2) - A_2 x_2}{A_1}$$

5. Step 5 — Evaluate:

   $$x_{1} = 3.7100\ \text{in}$$

6. Step 6 — Check: returning x_1 = 3.7100 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 7.4200 — kept a factor of two that cancels in the correct rearrangement.
• 1.8550 — dropped that same factor in the other direction.
• 4.0810 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

A machined bracket's area is decomposed into simple shapes to compute the centroid. Given area 1 (A_1) = 4.5000 in^2; centroid 1 x (x_1) = 2.4000 in; area 2 (A_2) = 19.0000 in^2; composite centroid x (x_bar) = 9.9200 in, determine the centroid 2 x (x_2) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• Everything except x_2 is given, so isolate x_2 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_2:

   $$x_{2} = \dfrac{\bar{x}(A_1+A_2) - A_1 x_1}{A_2}$$

3. Step 3 — List the givens: area 1 (A_1) = 4.5000 in^2, centroid 1 x (x_1) = 2.4000 in, area 2 (A_2) = 19.0000 in^2, composite centroid x (x_bar) = 9.9200 in.

4. Step 4 — Substitute the given values:

   $$x_{2} = \dfrac{\bar{x}(A_1+A_2) - A_1 x_1}{A_2}$$

5. Step 5 — Evaluate:

   $$x_{2} = 11.7011\ \text{in}$$

6. Step 6 — Check: returning x_2 = 11.7011 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 23.4021 — kept a factor of two that cancels in the correct rearrangement.
• 5.8505 — dropped that same factor in the other direction.
• 12.8712 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes

An angle bracket made of two rectangular plates needs its centroid located. Given area 1 (A_1) = 4.5000 in^2; centroid 1 x (x_1) = 9.3000 in; area 2 (A_2) = 9.0000 in^2; centroid 2 x (x_2) = 6.7000 in, determine the composite centroid x (x_bar) in in.

Start with the thinking

• The governing relation printed in this handbook section is Composite centroid (areas).
• 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.
• Locating the centroid of masses, areas, lengths, and volumes for a composite shape by weighting each part's area and centroid.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

2. Step 2 — Rearrange symbolically for x_bar:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

3. Step 3 — List the givens: area 1 (A_1) = 4.5000 in^2, centroid 1 x (x_1) = 9.3000 in, area 2 (A_2) = 9.0000 in^2, centroid 2 x (x_2) = 6.7000 in.

4. Step 4 — Substitute the given values:

   $$x_{bar} = \dfrac{A_1 x_1 + A_2 x_2}{A_1 + A_2}$$

5. Step 5 — Evaluate:

   $$x_{bar} = 7.5667\ \text{in}$$

6. Step 6 — Check: returning x_bar = 7.5667 in to

   $$\bar{x} = \dfrac{\sum A_i x_i}{\sum A_i}$$

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

Why the other options are there

• 15.1333 — kept a factor of two that cancels in the correct rearrangement.
• 3.7833 — dropped that same factor in the other direction.
• 8.3233 — rounded an intermediate value before the final step.

Reference: FE Handbook — Statics: Centroids of Masses, Areas, Lengths, and Volumes