population projection for a city's water treatment plant design year Given initial population (P_0) = 316,100; growth rate constant (k) = 0.0190 1/yr; time (t) = 29.0000 yr, determine the projected population (P_t).

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• Everything except P_t is given, so isolate P_t 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for P_t:

   $$P_{t} = P_0 e^{kt}$$

3. Step 3 — List the givens: initial population (P_0) = 316,100, growth rate constant (k) = 0.0190 1/yr, time (t) = 29.0000 yr.

4. Step 4 — Substitute the given values:

   $$P_{t} = P_0 e^{k29.0000}$$

5. Step 5 — Evaluate:

   $$P_{t} = 548429$$

6. Step 6 — Check: returning P_t = 548,429 to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 1,096,859 — kept a factor of two that cancels in the correct rearrangement.
• 274,215 — dropped that same factor in the other direction.
• 603,272 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

exponential population projection equation for a service area Given growth rate constant (k) = 0.0440 1/yr; time (t) = 31.0000 yr; projected population (P_t) = 1,294,109, determine the initial population (P_0).

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• Everything except P_0 is given, so isolate P_0 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for P_0:

   $$P_{0} = \dfrac{P_t}{e^{kt}}$$

3. Step 3 — List the givens: growth rate constant (k) = 0.0440 1/yr, time (t) = 31.0000 yr, projected population (P_t) = 1,294,109.

4. Step 4 — Substitute the given values:

   $$P_{0} = \dfrac{P_31.0000}{e^{k31.0000}}$$

5. Step 5 — Evaluate:

   $$P_{0} = 330821$$

6. Step 6 — Check: returning P_0 = 330,821 to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 661,642 — kept a factor of two that cancels in the correct rearrangement.
• 165,411 — dropped that same factor in the other direction.
• 363,903 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

population projection used to size a future wastewater treatment facility Given initial population (P_0) = 306,500; time (t) = 27.0000 yr; projected population (P_t) = 2,985,227, determine the growth rate constant (k) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \dfrac{\ln(P_t/P_0)}{t}$$

3. Step 3 — List the givens: initial population (P_0) = 306,500, time (t) = 27.0000 yr, projected population (P_t) = 2,985,227.

4. Step 4 — Substitute the given values:

   $$k = \dfrac{\ln(P_27.0000/P_0)}{27.0000}$$

5. Step 5 — Evaluate:

   $$k = 0.0843\ \text{1/yr}$$

6. Step 6 — Check: returning k = 0.0843 1/yr to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 0.1686 — kept a factor of two that cancels in the correct rearrangement.
• 0.0422 — dropped that same factor in the other direction.
• 0.0927 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

population projection for a city's water treatment plant design year Given initial population (P_0) = 145,300; growth rate constant (k) = 0.0270 1/yr; time (t) = 14.0000 yr, determine the projected population (P_t).

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• Everything except P_t is given, so isolate P_t 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for P_t:

   $$P_{t} = P_0 e^{kt}$$

3. Step 3 — List the givens: initial population (P_0) = 145,300, growth rate constant (k) = 0.0270 1/yr, time (t) = 14.0000 yr.

4. Step 4 — Substitute the given values:

   $$P_{t} = P_0 e^{k14.0000}$$

5. Step 5 — Evaluate:

   $$P_{t} = 212045$$

6. Step 6 — Check: returning P_t = 212,045 to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 424,091 — kept a factor of two that cancels in the correct rearrangement.
• 106,023 — dropped that same factor in the other direction.
• 233,250 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

exponential population projection equation for a service area Given growth rate constant (k) = 0.0210 1/yr; time (t) = 7.0000 yr; projected population (P_t) = 989,571, determine the initial population (P_0).

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• Everything except P_0 is given, so isolate P_0 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for P_0:

   $$P_{0} = \dfrac{P_t}{e^{kt}}$$

3. Step 3 — List the givens: growth rate constant (k) = 0.0210 1/yr, time (t) = 7.0000 yr, projected population (P_t) = 989,571.

4. Step 4 — Substitute the given values:

   $$P_{0} = \dfrac{P_7.0000}{e^{k7.0000}}$$

5. Step 5 — Evaluate:

   $$P_{0} = 854291$$

6. Step 6 — Check: returning P_0 = 854,291 to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 1,708,581 — kept a factor of two that cancels in the correct rearrangement.
• 427,145 — dropped that same factor in the other direction.
• 939,720 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

population projection used to size a future wastewater treatment facility Given initial population (P_0) = 214,700; time (t) = 31.0000 yr; projected population (P_t) = 375,518, determine the growth rate constant (k) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \dfrac{\ln(P_t/P_0)}{t}$$

3. Step 3 — List the givens: initial population (P_0) = 214,700, time (t) = 31.0000 yr, projected population (P_t) = 375,518.

4. Step 4 — Substitute the given values:

   $$k = \dfrac{\ln(P_31.0000/P_0)}{31.0000}$$

5. Step 5 — Evaluate:

   $$k = 0.0180\ \text{1/yr}$$

6. Step 6 — Check: returning k = 0.0180 1/yr to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 0.0361 — kept a factor of two that cancels in the correct rearrangement.
• 0.0090 — dropped that same factor in the other direction.
• 0.0198 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

population projection for a city's water treatment plant design year Given initial population (P_0) = 367,200; growth rate constant (k) = 0.0460 1/yr; time (t) = 16.0000 yr, determine the projected population (P_t).

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• Everything except P_t is given, so isolate P_t 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for P_t:

   $$P_{t} = P_0 e^{kt}$$

3. Step 3 — List the givens: initial population (P_0) = 367,200, growth rate constant (k) = 0.0460 1/yr, time (t) = 16.0000 yr.

4. Step 4 — Substitute the given values:

   $$P_{t} = P_0 e^{k16.0000}$$

5. Step 5 — Evaluate:

   $$P_{t} = 766555$$

6. Step 6 — Check: returning P_t = 766,555 to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 1,533,110 — kept a factor of two that cancels in the correct rearrangement.
• 383,278 — dropped that same factor in the other direction.
• 843,211 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

exponential population projection equation for a service area Given growth rate constant (k) = 0.0140 1/yr; time (t) = 23.0000 yr; projected population (P_t) = 2,760,134, determine the initial population (P_0).

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• Everything except P_0 is given, so isolate P_0 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for P_0:

   $$P_{0} = \dfrac{P_t}{e^{kt}}$$

3. Step 3 — List the givens: growth rate constant (k) = 0.0140 1/yr, time (t) = 23.0000 yr, projected population (P_t) = 2,760,134.

4. Step 4 — Substitute the given values:

   $$P_{0} = \dfrac{P_23.0000}{e^{k23.0000}}$$

5. Step 5 — Evaluate:

   $$P_{0} = 2000264$$

6. Step 6 — Check: returning P_0 = 2,000,264 to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 4,000,528 — kept a factor of two that cancels in the correct rearrangement.
• 1,000,132 — dropped that same factor in the other direction.
• 2,200,291 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

population projection used to size a future wastewater treatment facility Given initial population (P_0) = 419,700; time (t) = 1.0000 yr; projected population (P_t) = 1,647,525, determine the growth rate constant (k) in 1/yr.

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for k:

   $$k = \dfrac{\ln(P_t/P_0)}{t}$$

3. Step 3 — List the givens: initial population (P_0) = 419,700, time (t) = 1.0000 yr, projected population (P_t) = 1,647,525.

4. Step 4 — Substitute the given values:

   $$k = \dfrac{\ln(P_1.0000/P_0)}{1.0000}$$

5. Step 5 — Evaluate:

   $$k = 1.3675\ \text{1/yr}$$

6. Step 6 — Check: returning k = 1.3675 1/yr to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 2.7350 — kept a factor of two that cancels in the correct rearrangement.
• 0.6837 — dropped that same factor in the other direction.
• 1.5042 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations

population projection for a city's water treatment plant design year Given initial population (P_0) = 371,300; growth rate constant (k) = 0.0110 1/yr; time (t) = 35.0000 yr, determine the projected population (P_t).

Start with the thinking

• The governing relation printed in this handbook section is Population Projection Equations.
• Everything except P_t is given, so isolate P_t 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.
• Population projection equations use exponential growth to forecast future service population for water and wastewater planning.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$P_t = P_0 e^{k t}$$

2. Step 2 — Rearrange symbolically for P_t:

   $$P_{t} = P_0 e^{kt}$$

3. Step 3 — List the givens: initial population (P_0) = 371,300, growth rate constant (k) = 0.0110 1/yr, time (t) = 35.0000 yr.

4. Step 4 — Substitute the given values:

   $$P_{t} = P_0 e^{k35.0000}$$

5. Step 5 — Evaluate:

   $$P_{t} = 545668$$

6. Step 6 — Check: returning P_t = 545,668 to

   $$P_t = P_0 e^{k t}$$

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

Why the other options are there

• 1,091,336 — kept a factor of two that cancels in the correct rearrangement.
• 272,834 — dropped that same factor in the other direction.
• 600,235 — rounded an intermediate value before the final step.

Reference: FE Handbook — Population Projection Equations