a pendent sprinkler on a warehouse branch line Given sprinkler K factor (K) = 10.3000 gpm/psi^0.5; residual pressure at the head (P) = 63.0000 psi, determine the discharge (Q) in gpm.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except Q is given, so isolate Q 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = K \sqrt{P}$$

3. Step 3 — List the givens: sprinkler K factor (K) = 10.3000 gpm/psi^0.5, residual pressure at the head (P) = 63.0000 psi.

4. Step 4 — Substitute the given values:

   $$Q = 10.3000 \sqrt{63.0000}$$

5. Step 5 — Evaluate:

   $$Q = 81.7537\ \text{gpm}$$

6. Step 6 — Check: returning Q = 81.7537 gpm to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 163.5 — kept a factor of two that cancels in the correct rearrangement.
• 40.8769 — dropped that same factor in the other direction.
• 89.9291 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

an upright sprinkler protecting a parking structure Given discharge (Q) = 106.6 gpm; residual pressure at the head (P) = 37.5000 psi, determine the sprinkler K factor (K) in gpm/psi^0.5.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except K is given, so isolate K 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for K:

   $$K = \dfrac{Q}{\sqrt{P}}$$

3. Step 3

   $$List the givens: discharge (Q) = 106.6 gpm, residual pressure at the head (P) = 37.5000 psi$$

4. Step 4 — Substitute the given values:

   $$K = \dfrac{106.6}{\sqrt{37.5000}}$$

5. Step 5 — Evaluate:

   K = 17.4077\ \text{gpm/psi^0.5}

6. Step 6 — Check: returning K = 17.4077 gpm/psi^0.5 to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 34.8154 — kept a factor of two that cancels in the correct rearrangement.
• 8.7039 — dropped that same factor in the other direction.
• 19.1485 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

a sidewall sprinkler on a corridor main Given discharge (Q) = 134.9 gpm; sprinkler K factor (K) = 13.5000 gpm/psi^0.5, determine the residual pressure at the head (P) in psi.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except P is given, so isolate P 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for P:

   $$P = \left(\dfrac{Q}{K}\right)^2$$

3. Step 3

   $$List the givens: discharge (Q) = 134.9 gpm, sprinkler K factor (K) = 13.5000 gpm/psi^0.5$$

4. Step 4 — Substitute the given values:

   $$P = \left(\dfrac{134.9}{13.5000}\right)^2$$

5. Step 5 — Evaluate:

   $$P = 99.8519\ \text{psi}$$

6. Step 6 — Check: returning P = 99.8519 psi to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 199.7 — kept a factor of two that cancels in the correct rearrangement.
• 49.9260 — dropped that same factor in the other direction.
• 109.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

a pendent sprinkler on a warehouse branch line Given sprinkler K factor (K) = 5.7000 gpm/psi^0.5; residual pressure at the head (P) = 18.5000 psi, determine the discharge (Q) in gpm.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except Q is given, so isolate Q 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = K \sqrt{P}$$

3. Step 3 — List the givens: sprinkler K factor (K) = 5.7000 gpm/psi^0.5, residual pressure at the head (P) = 18.5000 psi.

4. Step 4 — Substitute the given values:

   $$Q = 5.7000 \sqrt{18.5000}$$

5. Step 5 — Evaluate:

   $$Q = 24.5166\ \text{gpm}$$

6. Step 6 — Check: returning Q = 24.5166 gpm to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 49.0333 — kept a factor of two that cancels in the correct rearrangement.
• 12.2583 — dropped that same factor in the other direction.
• 26.9683 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

an upright sprinkler protecting a parking structure Given discharge (Q) = 181.5 gpm; residual pressure at the head (P) = 67.5000 psi, determine the sprinkler K factor (K) in gpm/psi^0.5.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except K is given, so isolate K 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for K:

   $$K = \dfrac{Q}{\sqrt{P}}$$

3. Step 3

   $$List the givens: discharge (Q) = 181.5 gpm, residual pressure at the head (P) = 67.5000 psi$$

4. Step 4 — Substitute the given values:

   $$K = \dfrac{181.5}{\sqrt{67.5000}}$$

5. Step 5 — Evaluate:

   K = 22.0915\ \text{gpm/psi^0.5}

6. Step 6 — Check: returning K = 22.0915 gpm/psi^0.5 to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 44.1830 — kept a factor of two that cancels in the correct rearrangement.
• 11.0457 — dropped that same factor in the other direction.
• 24.3006 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

a sidewall sprinkler on a corridor main Given discharge (Q) = 37.7000 gpm; sprinkler K factor (K) = 6.2000 gpm/psi^0.5, determine the residual pressure at the head (P) in psi.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except P is given, so isolate P 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for P:

   $$P = \left(\dfrac{Q}{K}\right)^2$$

3. Step 3

   $$List the givens: discharge (Q) = 37.7000 gpm, sprinkler K factor (K) = 6.2000 gpm/psi^0.5$$

4. Step 4 — Substitute the given values:

   $$P = \left(\dfrac{37.7000}{6.2000}\right)^2$$

5. Step 5 — Evaluate:

   $$P = 36.9742\ \text{psi}$$

6. Step 6 — Check: returning P = 36.9742 psi to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 73.9485 — kept a factor of two that cancels in the correct rearrangement.
• 18.4871 — dropped that same factor in the other direction.
• 40.6717 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

a pendent sprinkler on a warehouse branch line Given sprinkler K factor (K) = 6.3000 gpm/psi^0.5; residual pressure at the head (P) = 88.5000 psi, determine the discharge (Q) in gpm.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except Q is given, so isolate Q 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = K \sqrt{P}$$

3. Step 3 — List the givens: sprinkler K factor (K) = 6.3000 gpm/psi^0.5, residual pressure at the head (P) = 88.5000 psi.

4. Step 4 — Substitute the given values:

   $$Q = 6.3000 \sqrt{88.5000}$$

5. Step 5 — Evaluate:

   $$Q = 59.2669\ \text{gpm}$$

6. Step 6 — Check: returning Q = 59.2669 gpm to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 118.5 — kept a factor of two that cancels in the correct rearrangement.
• 29.6334 — dropped that same factor in the other direction.
• 65.1936 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

an upright sprinkler protecting a parking structure Given discharge (Q) = 178.9 gpm; residual pressure at the head (P) = 53.5000 psi, determine the sprinkler K factor (K) in gpm/psi^0.5.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except K is given, so isolate K 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for K:

   $$K = \dfrac{Q}{\sqrt{P}}$$

3. Step 3

   $$List the givens: discharge (Q) = 178.9 gpm, residual pressure at the head (P) = 53.5000 psi$$

4. Step 4 — Substitute the given values:

   $$K = \dfrac{178.9}{\sqrt{53.5000}}$$

5. Step 5 — Evaluate:

   K = 24.4587\ \text{gpm/psi^0.5}

6. Step 6 — Check: returning K = 24.4587 gpm/psi^0.5 to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 48.9174 — kept a factor of two that cancels in the correct rearrangement.
• 12.2294 — dropped that same factor in the other direction.
• 26.9046 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

a sidewall sprinkler on a corridor main Given discharge (Q) = 98.3000 gpm; sprinkler K factor (K) = 8.6000 gpm/psi^0.5, determine the residual pressure at the head (P) in psi.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except P is given, so isolate P 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for P:

   $$P = \left(\dfrac{Q}{K}\right)^2$$

3. Step 3

   $$List the givens: discharge (Q) = 98.3000 gpm, sprinkler K factor (K) = 8.6000 gpm/psi^0.5$$

4. Step 4 — Substitute the given values:

   $$P = \left(\dfrac{98.3000}{8.6000}\right)^2$$

5. Step 5 — Evaluate:

   $$P = 130.7\ \text{psi}$$

6. Step 6 — Check: returning P = 130.7 psi to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 261.3 — kept a factor of two that cancels in the correct rearrangement.
• 65.3251 — dropped that same factor in the other direction.
• 143.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors

a pendent sprinkler on a warehouse branch line Given sprinkler K factor (K) = 12.7000 gpm/psi^0.5; residual pressure at the head (P) = 14.0000 psi, determine the discharge (Q) in gpm.

Start with the thinking

• The governing relation printed in this handbook section is Sprinkler discharge from the K factor.
• Everything except Q is given, so isolate Q 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.
• A fire-protection sprinkler head discharges under the residual pressure at the head.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$Q = K \sqrt{P}$$

2. Step 2 — Rearrange symbolically for Q:

   $$Q = K \sqrt{P}$$

3. Step 3 — List the givens: sprinkler K factor (K) = 12.7000 gpm/psi^0.5, residual pressure at the head (P) = 14.0000 psi.

4. Step 4 — Substitute the given values:

   $$Q = 12.7000 \sqrt{14.0000}$$

5. Step 5 — Evaluate:

   $$Q = 47.5190\ \text{gpm}$$

6. Step 6 — Check: returning Q = 47.5190 gpm to

   $$Q = K \sqrt{P}$$

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

Why the other options are there

• 95.0381 — kept a factor of two that cancels in the correct rearrangement.
• 23.7595 — dropped that same factor in the other direction.
• 52.2710 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sprinkler K Factors