An engineer applies the Laplace transform to an exponential decay response. Given decay constant (a) = 5.7000 1/s; complex frequency (s) = 5.8000 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

   $$List the givens: decay constant (a) = 5.7000 1/s, complex frequency (s) = 5.8000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{5.8000+5.7000}$$

5. Step 5 — Evaluate:

   $$F = 0.0870\ \text{s}$$

6. Step 6 — Check: returning F = 0.0870 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

• 0.1739 — kept a factor of two that cancels in the correct rearrangement.
• 0.0435 — dropped that same factor in the other direction.
• 0.0957 — 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) = 6.9000 1/s; transform value (F) = 0.1050 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

   $$List the givens: decay constant (a) = 6.9000 1/s, transform value (F) = 0.1050 s$$

4. Step 4 — Substitute the given values:

   $$s = \dfrac{1}{0.1050} - 6.9000$$

5. Step 5 — Evaluate:

   $$s = 2.6238\ \text{1/s}$$

6. Step 6 — Check: returning s = 2.6238 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

• 5.2476 — kept a factor of two that cancels in the correct rearrangement.
• 1.3119 — dropped that same factor in the other direction.
• 2.8862 — 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) = 0.7000 1/s; transform value (F) = 0.5800 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

   $$List the givens: complex frequency (s) = 0.7000 1/s, transform value (F) = 0.5800 s$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{1}{0.5800} - 0.7000$$

5. Step 5 — Evaluate:

   $$a = 1.0241\ \text{1/s}$$

6. Step 6 — Check: returning a = 1.0241 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

• 2.0483 — kept a factor of two that cancels in the correct rearrangement.
• 0.5121 — dropped that same factor in the other direction.
• 1.1266 — 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) = 5.6000 1/s; complex frequency (s) = 8.9000 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

   $$List the givens: decay constant (a) = 5.6000 1/s, complex frequency (s) = 8.9000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{8.9000+5.6000}$$

5. Step 5 — Evaluate:

   $$F = 0.0690\ \text{s}$$

6. Step 6 — Check: returning F = 0.0690 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

• 0.1379 — kept a factor of two that cancels in the correct rearrangement.
• 0.0345 — dropped that same factor in the other direction.
• 0.0759 — 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) = 5.6000 1/s; transform value (F) = 1.2750 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

   $$List the givens: decay constant (a) = 5.6000 1/s, transform value (F) = 1.2750 s$$

4. Step 4 — Substitute the given values:

   $$s = \dfrac{1}{1.2750} - 5.6000$$

5. Step 5 — Evaluate:

   $$s = -4.8157\ \text{1/s}$$

6. Step 6 — Check: returning s = -4.8157 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

• -9.6314 — kept a factor of two that cancels in the correct rearrangement.
• -2.4078 — dropped that same factor in the other direction.
• -5.2973 — 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) = 2.8000 1/s; transform value (F) = 1.1150 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

   $$List the givens: complex frequency (s) = 2.8000 1/s, transform value (F) = 1.1150 s$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{1}{1.1150} - 2.8000$$

5. Step 5 — Evaluate:

   $$a = -1.9031\ \text{1/s}$$

6. Step 6 — Check: returning a = -1.9031 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

• -3.8063 — kept a factor of two that cancels in the correct rearrangement.
• -0.9516 — dropped that same factor in the other direction.
• -2.0935 — 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) = 2.7000 1/s; complex frequency (s) = 2.0000 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

   $$List the givens: decay constant (a) = 2.7000 1/s, complex frequency (s) = 2.0000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{2.0000+2.7000}$$

5. Step 5 — Evaluate:

   $$F = 0.2128\ \text{s}$$

6. Step 6 — Check: returning F = 0.2128 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

• 0.4255 — kept a factor of two that cancels in the correct rearrangement.
• 0.1064 — dropped that same factor in the other direction.
• 0.2340 — 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) = 2.6000 1/s; transform value (F) = 0.4310 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

   $$List the givens: decay constant (a) = 2.6000 1/s, transform value (F) = 0.4310 s$$

4. Step 4 — Substitute the given values:

   $$s = \dfrac{1}{0.4310} - 2.6000$$

5. Step 5 — Evaluate:

   $$s = -0.2798\ \text{1/s}$$

6. Step 6 — Check: returning s = -0.2798 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

• -0.5596 — kept a factor of two that cancels in the correct rearrangement.
• -0.1399 — dropped that same factor in the other direction.
• -0.3078 — 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) = 10.0000 1/s; transform value (F) = 0.6200 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

   $$List the givens: complex frequency (s) = 10.0000 1/s, transform value (F) = 0.6200 s$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{1}{0.6200} - 10.0000$$

5. Step 5 — Evaluate:

   $$a = -8.3871\ \text{1/s}$$

6. Step 6 — Check: returning a = -8.3871 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

• -16.7742 — kept a factor of two that cancels in the correct rearrangement.
• -4.1935 — dropped that same factor in the other direction.
• -9.2258 — 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) = 6.7000 1/s; complex frequency (s) = 9.8000 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

   $$List the givens: decay constant (a) = 6.7000 1/s, complex frequency (s) = 9.8000 1/s$$

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1}{9.8000+6.7000}$$

5. Step 5 — Evaluate:

   $$F = 0.0606\ \text{s}$$

6. Step 6 — Check: returning F = 0.0606 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

• 0.1212 — kept a factor of two that cancels in the correct rearrangement.
• 0.0303 — dropped that same factor in the other direction.
• 0.0667 — rounded an intermediate value before the final step.

Reference: FE Handbook — Laplace Transforms