compaction ratio of solid waste in a landfill cell Given compacted unit weight (gamma_c) = 495.0 kg/m^3; loose unit weight (gamma_l) = 200.0 kg/m^3, determine the compaction ratio (CR).

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except CR is given, so isolate CR 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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for CR:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

3. Step 3 — List the givens: compacted unit weight (gamma_c) = 495.0 kg/m^3, loose unit weight (gamma_l) = 200.0 kg/m^3.

4. Step 4 — Substitute the given values:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

5. Step 5 — Evaluate:

   $$CR = 2.4750$$

6. Step 6 — Check: returning CR = 2.4750 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 4.9500 — kept a factor of two that cancels in the correct rearrangement.
• 1.2375 — dropped that same factor in the other direction.
• 2.7225 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

compaction of municipal solid waste by a landfill compactor Given loose unit weight (gamma_l) = 295.0 kg/m^3; compaction ratio (CR) = 4.6000, determine the compacted unit weight (gamma_c) in kg/m^3.

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except gamma_c is given, so isolate gamma_c 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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for gamma_c:

   $$\gamma_{c} = CR\,\gamma_{loose}$$

3. Step 3 — List the givens: loose unit weight (gamma_l) = 295.0 kg/m^3, compaction ratio (CR) = 4.6000.

4. Step 4 — Substitute the given values:

   $$\gamma_{c} = 4.6000\,\gamma_{loose}$$

5. Step 5 — Evaluate:

   \gamma_{c} = 1357\ \text{kg/m^3}

6. Step 6 — Check: returning gamma_c = 1,357 kg/m^3 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 2,714 — kept a factor of two that cancels in the correct rearrangement.
• 678.5 — dropped that same factor in the other direction.
• 1,493 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

waste compaction ratio used to estimate landfill airspace consumption Given compacted unit weight (gamma_c) = 910.0 kg/m^3; compaction ratio (CR) = 4.7500, determine the loose unit weight (gamma_l) in kg/m^3.

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except gamma_l is given, so isolate gamma_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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for gamma_l:

   $$\gamma_{l} = \dfrac{\gamma_{compacted}}{CR}$$

3. Step 3 — List the givens: compacted unit weight (gamma_c) = 910.0 kg/m^3, compaction ratio (CR) = 4.7500.

4. Step 4 — Substitute the given values:

   $$\gamma_{l} = \dfrac{\gamma_{compacted}}{4.7500}$$

5. Step 5 — Evaluate:

   \gamma_{l} = 191.6\ \text{kg/m^3}

6. Step 6 — Check: returning gamma_l = 191.6 kg/m^3 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 383.2 — kept a factor of two that cancels in the correct rearrangement.
• 95.7895 — dropped that same factor in the other direction.
• 210.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

compaction ratio of solid waste in a landfill cell Given compacted unit weight (gamma_c) = 930.0 kg/m^3; loose unit weight (gamma_l) = 250.0 kg/m^3, determine the compaction ratio (CR).

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except CR is given, so isolate CR 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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for CR:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

3. Step 3 — List the givens: compacted unit weight (gamma_c) = 930.0 kg/m^3, loose unit weight (gamma_l) = 250.0 kg/m^3.

4. Step 4 — Substitute the given values:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

5. Step 5 — Evaluate:

   $$CR = 3.7200$$

6. Step 6 — Check: returning CR = 3.7200 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 7.4400 — kept a factor of two that cancels in the correct rearrangement.
• 1.8600 — dropped that same factor in the other direction.
• 4.0920 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

compaction of municipal solid waste by a landfill compactor Given loose unit weight (gamma_l) = 200.0 kg/m^3; compaction ratio (CR) = 4.2500, determine the compacted unit weight (gamma_c) in kg/m^3.

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except gamma_c is given, so isolate gamma_c 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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for gamma_c:

   $$\gamma_{c} = CR\,\gamma_{loose}$$

3. Step 3 — List the givens: loose unit weight (gamma_l) = 200.0 kg/m^3, compaction ratio (CR) = 4.2500.

4. Step 4 — Substitute the given values:

   $$\gamma_{c} = 4.2500\,\gamma_{loose}$$

5. Step 5 — Evaluate:

   \gamma_{c} = 850.0\ \text{kg/m^3}

6. Step 6 — Check: returning gamma_c = 850.0 kg/m^3 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 1,700 — kept a factor of two that cancels in the correct rearrangement.
• 425.0 — dropped that same factor in the other direction.
• 935.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

waste compaction ratio used to estimate landfill airspace consumption Given compacted unit weight (gamma_c) = 515.0 kg/m^3; compaction ratio (CR) = 2.6500, determine the loose unit weight (gamma_l) in kg/m^3.

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except gamma_l is given, so isolate gamma_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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for gamma_l:

   $$\gamma_{l} = \dfrac{\gamma_{compacted}}{CR}$$

3. Step 3 — List the givens: compacted unit weight (gamma_c) = 515.0 kg/m^3, compaction ratio (CR) = 2.6500.

4. Step 4 — Substitute the given values:

   $$\gamma_{l} = \dfrac{\gamma_{compacted}}{2.6500}$$

5. Step 5 — Evaluate:

   \gamma_{l} = 194.3\ \text{kg/m^3}

6. Step 6 — Check: returning gamma_l = 194.3 kg/m^3 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 388.7 — kept a factor of two that cancels in the correct rearrangement.
• 97.1698 — dropped that same factor in the other direction.
• 213.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

compaction ratio of solid waste in a landfill cell Given compacted unit weight (gamma_c) = 870.0 kg/m^3; loose unit weight (gamma_l) = 135.0 kg/m^3, determine the compaction ratio (CR).

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except CR is given, so isolate CR 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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for CR:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

3. Step 3 — List the givens: compacted unit weight (gamma_c) = 870.0 kg/m^3, loose unit weight (gamma_l) = 135.0 kg/m^3.

4. Step 4 — Substitute the given values:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

5. Step 5 — Evaluate:

   $$CR = 6.4444$$

6. Step 6 — Check: returning CR = 6.4444 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 12.8889 — kept a factor of two that cancels in the correct rearrangement.
• 3.2222 — dropped that same factor in the other direction.
• 7.0889 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

compaction of municipal solid waste by a landfill compactor Given loose unit weight (gamma_l) = 355.0 kg/m^3; compaction ratio (CR) = 4.9000, determine the compacted unit weight (gamma_c) in kg/m^3.

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except gamma_c is given, so isolate gamma_c 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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for gamma_c:

   $$\gamma_{c} = CR\,\gamma_{loose}$$

3. Step 3 — List the givens: loose unit weight (gamma_l) = 355.0 kg/m^3, compaction ratio (CR) = 4.9000.

4. Step 4 — Substitute the given values:

   $$\gamma_{c} = 4.9000\,\gamma_{loose}$$

5. Step 5 — Evaluate:

   \gamma_{c} = 1740\ \text{kg/m^3}

6. Step 6 — Check: returning gamma_c = 1,740 kg/m^3 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 3,479 — kept a factor of two that cancels in the correct rearrangement.
• 869.8 — dropped that same factor in the other direction.
• 1,913 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

waste compaction ratio used to estimate landfill airspace consumption Given compacted unit weight (gamma_c) = 755.0 kg/m^3; compaction ratio (CR) = 4.8500, determine the loose unit weight (gamma_l) in kg/m^3.

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except gamma_l is given, so isolate gamma_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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for gamma_l:

   $$\gamma_{l} = \dfrac{\gamma_{compacted}}{CR}$$

3. Step 3 — List the givens: compacted unit weight (gamma_c) = 755.0 kg/m^3, compaction ratio (CR) = 4.8500.

4. Step 4 — Substitute the given values:

   $$\gamma_{l} = \dfrac{\gamma_{compacted}}{4.8500}$$

5. Step 5 — Evaluate:

   \gamma_{l} = 155.7\ \text{kg/m^3}

6. Step 6 — Check: returning gamma_l = 155.7 kg/m^3 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 311.3 — kept a factor of two that cancels in the correct rearrangement.
• 77.8351 — dropped that same factor in the other direction.
• 171.2 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction

compaction ratio of solid waste in a landfill cell Given compacted unit weight (gamma_c) = 920.0 kg/m^3; loose unit weight (gamma_l) = 380.0 kg/m^3, determine the compaction ratio (CR).

Start with the thinking

• The governing relation printed in this handbook section is Compaction.
• Everything except CR is given, so isolate CR 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.
• Compaction of solid waste is characterized by the compaction ratio comparing compacted density to loose density in a landfill.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

2. Step 2 — Rearrange symbolically for CR:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

3. Step 3 — List the givens: compacted unit weight (gamma_c) = 920.0 kg/m^3, loose unit weight (gamma_l) = 380.0 kg/m^3.

4. Step 4 — Substitute the given values:

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

5. Step 5 — Evaluate:

   $$CR = 2.4211$$

6. Step 6 — Check: returning CR = 2.4211 to

   $$CR = \dfrac{\gamma_{compacted}}{\gamma_{loose}}$$

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

Why the other options are there

• 4.8421 — kept a factor of two that cancels in the correct rearrangement.
• 1.2105 — dropped that same factor in the other direction.
• 2.6632 — rounded an intermediate value before the final step.

Reference: FE Handbook — Compaction