An engineer applies the Laplace transform to an exponential decay response. Given decay constant (a) = 7.7000 1/s; complex frequency (s) = 8.5000 1/s, determine the transform value (F) in s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except F is given, so isolate F 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{1}{s+a}$$

3. Step 3

   $$List the givens: decay constant (a) = 7.7000 1/s, complex frequency (s) = 8.5000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{8.5000+7.7000}$$

5. Step 5 — Evaluate:

   $$F = 0.0617\ \text{s}$$

6. Step 6 — Check: returning F = 0.0617 s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• 0.1235 — kept a factor of two that cancels in the correct rearrangement.
• 0.0309 — dropped that same factor in the other direction.
• 0.0679 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

A student looks up a Laplace transform pair for a first-order system. Given decay constant (a) = 1.5000 1/s; transform value (F) = 1.7150 s, determine the complex frequency (s) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except s is given, so isolate s 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for s:

   $$s = \dfrac{1}{F} - a$$

3. Step 3

   $$List the givens: decay constant (a) = 1.5000 1/s, transform value (F) = 1.7150 s$$

4. Step 4 — Substitute the given values:

   $$s = \dfrac{1}{1.7150} - 1.5000$$

5. Step 5 — Evaluate:

   $$s = -0.9169\ \text{1/s}$$

6. Step 6 — Check: returning s = -0.9169 1/s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• -1.8338 — kept a factor of two that cancels in the correct rearrangement.
• -0.4585 — dropped that same factor in the other direction.
• -1.0086 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

The Laplace transform of a circuit's impulse response is evaluated at s. Given complex frequency (s) = 1.7000 1/s; transform value (F) = 1.0940 s, determine the decay constant (a) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except a is given, so isolate a 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{1}{F} - s$$

3. Step 3

   $$List the givens: complex frequency (s) = 1.7000 1/s, transform value (F) = 1.0940 s$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{1}{1.0940} - 1.7000$$

5. Step 5 — Evaluate:

   $$a = -0.7859\ \text{1/s}$$

6. Step 6 — Check: returning a = -0.7859 1/s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• -1.5718 — kept a factor of two that cancels in the correct rearrangement.
• -0.3930 — dropped that same factor in the other direction.
• -0.8645 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

An engineer applies the Laplace transform to an exponential decay response. Given decay constant (a) = 1.3000 1/s; complex frequency (s) = 3.8000 1/s, determine the transform value (F) in s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except F is given, so isolate F 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{1}{s+a}$$

3. Step 3

   $$List the givens: decay constant (a) = 1.3000 1/s, complex frequency (s) = 3.8000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{3.8000+1.3000}$$

5. Step 5 — Evaluate:

   $$F = 0.1961\ \text{s}$$

6. Step 6 — Check: returning F = 0.1961 s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• 0.3922 — kept a factor of two that cancels in the correct rearrangement.
• 0.0980 — dropped that same factor in the other direction.
• 0.2157 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

A student looks up a Laplace transform pair for a first-order system. Given decay constant (a) = 2.4000 1/s; transform value (F) = 1.8060 s, determine the complex frequency (s) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except s is given, so isolate s 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for s:

   $$s = \dfrac{1}{F} - a$$

3. Step 3

   $$List the givens: decay constant (a) = 2.4000 1/s, transform value (F) = 1.8060 s$$

4. Step 4 — Substitute the given values:

   $$s = \dfrac{1}{1.8060} - 2.4000$$

5. Step 5 — Evaluate:

   $$s = -1.8463\ \text{1/s}$$

6. Step 6 — Check: returning s = -1.8463 1/s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• -3.6926 — kept a factor of two that cancels in the correct rearrangement.
• -0.9231 — dropped that same factor in the other direction.
• -2.0309 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

The Laplace transform of a circuit's impulse response is evaluated at s. Given complex frequency (s) = 5.2000 1/s; transform value (F) = 0.9280 s, determine the decay constant (a) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except a is given, so isolate a 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{1}{F} - s$$

3. Step 3

   $$List the givens: complex frequency (s) = 5.2000 1/s, transform value (F) = 0.9280 s$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{1}{0.9280} - 5.2000$$

5. Step 5 — Evaluate:

   $$a = -4.1224\ \text{1/s}$$

6. Step 6 — Check: returning a = -4.1224 1/s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• -8.2448 — kept a factor of two that cancels in the correct rearrangement.
• -2.0612 — dropped that same factor in the other direction.
• -4.5347 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

An engineer applies the Laplace transform to an exponential decay response. Given decay constant (a) = 7.0000 1/s; complex frequency (s) = 0.6000 1/s, determine the transform value (F) in s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except F is given, so isolate F 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{1}{s+a}$$

3. Step 3

   $$List the givens: decay constant (a) = 7.0000 1/s, complex frequency (s) = 0.6000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{0.6000+7.0000}$$

5. Step 5 — Evaluate:

   $$F = 0.1316\ \text{s}$$

6. Step 6 — Check: returning F = 0.1316 s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• 0.2632 — kept a factor of two that cancels in the correct rearrangement.
• 0.0658 — dropped that same factor in the other direction.
• 0.1447 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

A student looks up a Laplace transform pair for a first-order system. Given decay constant (a) = 1.7000 1/s; transform value (F) = 0.8070 s, determine the complex frequency (s) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except s is given, so isolate s 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for s:

   $$s = \dfrac{1}{F} - a$$

3. Step 3

   $$List the givens: decay constant (a) = 1.7000 1/s, transform value (F) = 0.8070 s$$

4. Step 4 — Substitute the given values:

   $$s = \dfrac{1}{0.8070} - 1.7000$$

5. Step 5 — Evaluate:

   $$s = -0.4608\ \text{1/s}$$

6. Step 6 — Check: returning s = -0.4608 1/s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• -0.9217 — kept a factor of two that cancels in the correct rearrangement.
• -0.2304 — dropped that same factor in the other direction.
• -0.5069 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

The Laplace transform of a circuit's impulse response is evaluated at s. Given complex frequency (s) = 2.6000 1/s; transform value (F) = 1.6690 s, determine the decay constant (a) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except a is given, so isolate a 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{1}{F} - s$$

3. Step 3

   $$List the givens: complex frequency (s) = 2.6000 1/s, transform value (F) = 1.6690 s$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{1}{1.6690} - 2.6000$$

5. Step 5 — Evaluate:

   $$a = -2.0008\ \text{1/s}$$

6. Step 6 — Check: returning a = -2.0008 1/s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• -4.0017 — kept a factor of two that cancels in the correct rearrangement.
• -1.0004 — dropped that same factor in the other direction.
• -2.2009 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms

An engineer applies the Laplace transform to an exponential decay response. Given decay constant (a) = 8.9000 1/s; complex frequency (s) = 9.6000 1/s, determine the transform value (F) in s.

Start with the thinking

• The governing relation printed in this handbook section is Laplace transform of an exponential.
• Everything except F is given, so isolate F 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.
• The Laplace transform of e^{-at} is 1/(s+a), a standard entry used in transform tables.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$F(s) = \dfrac{1}{s + a}$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{1}{s+a}$$

3. Step 3

   $$List the givens: decay constant (a) = 8.9000 1/s, complex frequency (s) = 9.6000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{9.6000+8.9000}$$

5. Step 5 — Evaluate:

   $$F = 0.0541\ \text{s}$$

6. Step 6 — Check: returning F = 0.0541 s to

   $$F(s) = \dfrac{1}{s + a}$$

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

Why the other options are there

• 0.1081 — kept a factor of two that cancels in the correct rearrangement.
• 0.0270 — dropped that same factor in the other direction.
• 0.0595 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms