A mathematics problem uses Exponential decay. Given initial value (y0) = 23.5000; decay constant (k) = 1.9000 1/s; time (t) = 0.1500 s, determine the value at t (y).

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except y is given, so isolate y 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3 — List the givens: initial value (y0) = 23.5000, decay constant (k) = 1.9000 1/s, time (t) = 0.1500 s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 17.6723$$

6. Step 6 — Check: returning y = 17.6723 to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• 35.3447 — kept a factor of two that cancels in the correct rearrangement.
• 8.8362 — dropped that same factor in the other direction.
• 19.4396 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given decay constant (k) = 1.0000 1/s; time (t) = 2.2000 s; value at t (y) = 27.2800, determine the initial value (y0).

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except y0 is given, so isolate y0 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that y0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: decay constant (k) = 1.0000 1/s, time (t) = 2.2000 s, value at t (y) = 27.2800$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y_{0} = 246.2$$

6. Step 6 — Check: returning y0 = 246.2 to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

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

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given initial value (y0) = 28.0000; time (t) = 0.3000 s; value at t (y) = 31.5100, determine the decay constant (k) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3

   $$List the givens: initial value (y_{0}) = 28.0000, time (t) = 0.3000 s, value at t (y) = 31.5100$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = -0.3937\ \text{1/s}$$

6. Step 6 — Check: returning k = -0.3937 1/s to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• -0.7873 — kept a factor of two that cancels in the correct rearrangement.
• -0.1968 — dropped that same factor in the other direction.
• -0.4330 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given initial value (y0) = 48.0000; decay constant (k) = 1.9000 1/s; value at t (y) = 9.6100, determine the time (t) in s.

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except t is given, so isolate 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that t stands alone on the left-hand side.

3. Step 3 — List the givens: initial value (y0) = 48.0000, decay constant (k) = 1.9000 1/s, value at t (y) = 9.6100.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$t = 0.8465\ \text{s}$$

6. Step 6 — Check: returning t = 0.8465 s to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• 1.6930 — kept a factor of two that cancels in the correct rearrangement.
• 0.4233 — dropped that same factor in the other direction.
• 0.9312 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given initial value (y0) = 25.5000; decay constant (k) = 0.7500 1/s; time (t) = 0.9500 s, determine the value at t (y).

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except y is given, so isolate y 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3 — List the givens: initial value (y0) = 25.5000, decay constant (k) = 0.7500 1/s, time (t) = 0.9500 s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 12.5056$$

6. Step 6 — Check: returning y = 12.5056 to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• 25.0112 — kept a factor of two that cancels in the correct rearrangement.
• 6.2528 — dropped that same factor in the other direction.
• 13.7562 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given decay constant (k) = 0.6000 1/s; time (t) = 1.9000 s; value at t (y) = 35.3400, determine the initial value (y0).

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except y0 is given, so isolate y0 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that y0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: decay constant (k) = 0.6000 1/s, time (t) = 1.9000 s, value at t (y) = 35.3400$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y_{0} = 110.5$$

6. Step 6 — Check: returning y0 = 110.5 to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• 221.0 — kept a factor of two that cancels in the correct rearrangement.
• 55.2500 — dropped that same factor in the other direction.
• 121.5 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given initial value (y0) = 40.5000; time (t) = 1.9000 s; value at t (y) = 47.6700, determine the decay constant (k) in 1/s.

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that k stands alone on the left-hand side.

3. Step 3

   $$List the givens: initial value (y_{0}) = 40.5000, time (t) = 1.9000 s, value at t (y) = 47.6700$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$k = -0.0858\ \text{1/s}$$

6. Step 6 — Check: returning k = -0.0858 1/s to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• -0.1716 — kept a factor of two that cancels in the correct rearrangement.
• -0.0429 — dropped that same factor in the other direction.
• -0.0944 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given initial value (y0) = 9.0000; decay constant (k) = 1.6000 1/s; value at t (y) = 31.4500, determine the time (t) in s.

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except t is given, so isolate 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that t stands alone on the left-hand side.

3. Step 3 — List the givens: initial value (y0) = 9.0000, decay constant (k) = 1.6000 1/s, value at t (y) = 31.4500.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$t = -0.7820\ \text{s}$$

6. Step 6 — Check: returning t = -0.7820 s to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• -1.5640 — kept a factor of two that cancels in the correct rearrangement.
• -0.3910 — dropped that same factor in the other direction.
• -0.8602 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given initial value (y0) = 12.5000; decay constant (k) = 1.7500 1/s; time (t) = 2.5500 s, determine the value at t (y).

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except y is given, so isolate y 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that y stands alone on the left-hand side.

3. Step 3 — List the givens: initial value (y0) = 12.5000, decay constant (k) = 1.7500 1/s, time (t) = 2.5500 s.

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y = 0.1442$$

6. Step 6 — Check: returning y = 0.1442 to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• 0.2883 — kept a factor of two that cancels in the correct rearrangement.
• 0.0721 — dropped that same factor in the other direction.
• 0.1586 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus

A mathematics problem uses Exponential decay. Given decay constant (k) = 1.8500 1/s; time (t) = 0.2500 s; value at t (y) = 37.7400, determine the initial value (y0).

Start with the thinking

• The governing relation printed in this handbook section is Exponential decay.
• Everything except y0 is given, so isolate y0 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.
• Mathematics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = y_0 e^{-k t}$$

2. Step 2 — Rearrange the relation so that y0 stands alone on the left-hand side.

3. Step 3

   $$List the givens: decay constant (k) = 1.8500 1/s, time (t) = 0.2500 s, value at t (y) = 37.7400$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$y_{0} = 59.9326$$

6. Step 6 — Check: returning y0 = 59.9326 to

   $$y = y_0 e^{-k t}$$

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

Why the other options are there

• 119.9 — kept a factor of two that cancels in the correct rearrangement.
• 29.9663 — dropped that same factor in the other direction.
• 65.9259 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Differential Calculus