Two cross sections 30 m apart have cut areas of 24.5 m² and 31.7 m². Compute the volume between them, then the loose volume with 12% swell.

Start with the thinking

• Average end area is the FE default unless the prismoidal method is requested.
• Swell increases hauled volume relative to bank volume.

Step-by-step solution

1. Average area

   $$(24.5 + 31.7)/2 = 28.1 m^{2}$$

2. Bank volume

   $$V = 28.1(30) = 843 m^{3}$$

3. Swell factor

   $$1 + 0.12 = 1.12$$

4. Loose volume — 843(1.12)

5. Result

   $$V_bank = 843 m^{3}, V_loose = 944 m^{3}$$

Why the other options are there

• 1,686 m³ (areas added, not averaged)
• 753 m³ (shrinkage applied instead of swell)

Reference: FE Reference Handbook — Construction — Earthwork volumes

A surveying problem uses Average-end-area earthwork volume. Given station spacing (L) = 430.0 ft; end area 1 (A1) = 397.0 ft²; end area 2 (A2) = 35.0000 ft², determine the volume (V) in ft³.

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except V is given, so isolate V 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3

   $$List the givens: station spacing (L) = 430.0 ft, end area 1 (A_{1}) = 397.0 ft^{2}, end area 2 (A_{2}) = 35.0000 ft^{2}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 92880\ \text{ft³}$$

6. Step 6 — Check: returning V = 92,880 ft³ to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

• 185,760 — kept a factor of two that cancels in the correct rearrangement.
• 46,440 — dropped that same factor in the other direction.
• 102,168 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given end area 1 (A1) = 268.0 ft²; end area 2 (A2) = 46.0000 ft²; volume (V) = 148,838 ft³, determine the station spacing (L) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except L is given, so isolate L 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that L stands alone on the left-hand side.

3. Step 3

   $$List the givens: end area 1 (A_{1}) = 268.0 ft^{2}, end area 2 (A_{2}) = 46.0000 ft^{2}, volume (V) = 148,838 ft^{3}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L = 948.0\ \text{ft}$$

6. Step 6 — Check: returning L = 948.0 ft to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

• 1,896 — kept a factor of two that cancels in the correct rearrangement.
• 474.0 — dropped that same factor in the other direction.
• 1,043 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given station spacing (L) = 385.0 ft; end area 2 (A2) = 164.0 ft²; volume (V) = 129,048 ft³, determine the end area 1 (A1) in ft².

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except A1 is given, so isolate A1 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that A1 stands alone on the left-hand side.

3. Step 3

   $$List the givens: station spacing (L) = 385.0 ft, end area 2 (A_{2}) = 164.0 ft^{2}, volume (V) = 129,048 ft^{3}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$A_{1} = 506.4\ \text{ft²}$$

6. Step 6 — Check: returning A1 = 506.4 ft² to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

• 1,013 — kept a factor of two that cancels in the correct rearrangement.
• 253.2 — dropped that same factor in the other direction.
• 557.0 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given station spacing (L) = 210.0 ft; end area 1 (A1) = 260.0 ft²; end area 2 (A2) = 315.0 ft², determine the volume (V) in ft³.

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except V is given, so isolate V 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3

   $$List the givens: station spacing (L) = 210.0 ft, end area 1 (A_{1}) = 260.0 ft^{2}, end area 2 (A_{2}) = 315.0 ft^{2}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 60375\ \text{ft³}$$

6. Step 6 — Check: returning V = 60,375 ft³ to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

• 120,750 — kept a factor of two that cancels in the correct rearrangement.
• 30,188 — dropped that same factor in the other direction.
• 66,413 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given end area 1 (A1) = 120.0 ft²; end area 2 (A2) = 298.0 ft²; volume (V) = 100,553 ft³, determine the station spacing (L) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except L is given, so isolate L 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that L stands alone on the left-hand side.

3. Step 3

   $$List the givens: end area 1 (A_{1}) = 120.0 ft^{2}, end area 2 (A_{2}) = 298.0 ft^{2}, volume (V) = 100,553 ft^{3}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L = 481.1\ \text{ft}$$

6. Step 6 — Check: returning L = 481.1 ft to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

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

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given station spacing (L) = 285.0 ft; end area 2 (A2) = 343.0 ft²; volume (V) = 12,696 ft³, determine the end area 1 (A1) in ft².

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except A1 is given, so isolate A1 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that A1 stands alone on the left-hand side.

3. Step 3

   $$List the givens: station spacing (L) = 285.0 ft, end area 2 (A_{2}) = 343.0 ft^{2}, volume (V) = 12,696 ft^{3}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$A_{1} = -253.9\ \text{ft²}$$

6. Step 6 — Check: returning A1 = -253.9 ft² to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

• -507.8 — kept a factor of two that cancels in the correct rearrangement.
• -127.0 — dropped that same factor in the other direction.
• -279.3 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given station spacing (L) = 395.0 ft; end area 1 (A1) = 384.0 ft²; end area 2 (A2) = 245.0 ft², determine the volume (V) in ft³.

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except V is given, so isolate V 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that V stands alone on the left-hand side.

3. Step 3

   $$List the givens: station spacing (L) = 395.0 ft, end area 1 (A_{1}) = 384.0 ft^{2}, end area 2 (A_{2}) = 245.0 ft^{2}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$V = 124228\ \text{ft³}$$

6. Step 6 — Check: returning V = 124,228 ft³ to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

• 248,455 — kept a factor of two that cancels in the correct rearrangement.
• 62,114 — dropped that same factor in the other direction.
• 136,650 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given end area 1 (A1) = 304.0 ft²; end area 2 (A2) = 133.0 ft²; volume (V) = 155,798 ft³, determine the station spacing (L) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except L is given, so isolate L 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that L stands alone on the left-hand side.

3. Step 3

   $$List the givens: end area 1 (A_{1}) = 304.0 ft^{2}, end area 2 (A_{2}) = 133.0 ft^{2}, volume (V) = 155,798 ft^{3}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$L = 713.0\ \text{ft}$$

6. Step 6 — Check: returning L = 713.0 ft to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

• 1,426 — kept a factor of two that cancels in the correct rearrangement.
• 356.5 — dropped that same factor in the other direction.
• 784.3 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Surveying → Earthwork formulas

A surveying problem uses Average-end-area earthwork volume. Given station spacing (L) = 215.0 ft; end area 2 (A2) = 153.0 ft²; volume (V) = 55,937 ft³, determine the end area 1 (A1) in ft².

Start with the thinking

• The governing relation printed in this handbook section is Average-end-area earthwork volume.
• Everything except A1 is given, so isolate A1 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.
• Surveying items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = L (A_1 + A_2) / 2$$

2. Step 2 — Rearrange the relation so that A1 stands alone on the left-hand side.

3. Step 3

   $$List the givens: station spacing (L) = 215.0 ft, end area 2 (A_{2}) = 153.0 ft^{2}, volume (V) = 55,937 ft^{3}$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$A_{1} = 367.3\ \text{ft²}$$

6. Step 6 — Check: returning A1 = 367.3 ft² to

   $$V = L (A_1 + A_2) / 2$$

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

Why the other options are there

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

Reference: FE Reference Handbook — Surveying → Earthwork formulas