A mathematics problem uses Slope–intercept line. Given slope (m) = 4.6000; x-value (x) = 5.5000; intercept (b) = 1.0000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Slope–intercept line.
• 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:

   $$y = m x + b$$

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

3. Step 3

   $$List the givens: slope (m) = 4.6000, x-value (x) = 5.5000, intercept (b) = 1.0000$$

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

5. Step 5 — Evaluate:

   $$y = 26.3000$$

6. Step 6 — Check: returning y = 26.3000 to

   $$y = m x + b$$

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

Why the other options are there

• 52.6000 — kept a factor of two that cancels in the correct rearrangement.
• 13.1500 — dropped that same factor in the other direction.
• 28.9300 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Nonhomogeneous Differential Equations

A mathematics problem uses Exponential decay. Given initial value (y0) = 45.0000; decay constant (k) = 1.7500 1/s; time (t) = 0.8500 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) = 45.0000, decay constant (k) = 1.7500 1/s, time (t) = 0.8500 s.

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

5. Step 5 — Evaluate:

   $$y = 10.1672$$

6. Step 6 — Check: returning y = 10.1672 to

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

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

Why the other options are there

• 20.3343 — kept a factor of two that cancels in the correct rearrangement.
• 5.0836 — dropped that same factor in the other direction.
• 11.1839 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Nonhomogeneous Differential Equations

temperature decay of a mass concrete pour Given initial value (y_0) = 2.2000 mg/L; rate constant (k) = 0.7000 1/h; elapsed time (t) = 3.2000 h, determine the value at time t (y) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order linear ODE solution.
• 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.
• The ODE y' + k y = 0 has the exponential solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

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

2. Step 2 — Rearrange symbolically for y:

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

3. Step 3 — List the givens: initial value (y_0) = 2.2000 mg/L, rate constant (k) = 0.7000 1/h, elapsed time (t) = 3.2000 h.

4. Step 4 — Substitute the given values:

   $$y = y = y_0 e^{-0.7000 3.2000}$$

5. Step 5 — Evaluate:

   $$y = 0.2342\ \text{mg/L}$$

6. Step 6 — Check: returning y = 0.2342 mg/L to

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

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

Why the other options are there

• 0.4684 — kept a factor of two that cancels in the correct rearrangement.
• 0.1171 — dropped that same factor in the other direction.
• 0.2576 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (First Order)

m y'' + k y = 0 gives simple harmonic motion at \omega_n. Given stiffness (k) = 21,625 N/m; mass (m) = 295.8 kg, determine the natural frequency (\omega_n) in rad/s.

Start with the thinking

• The governing relation printed in this handbook section is Undamped natural frequency (2nd-order ODE).
• Everything except \omega_n is given, so isolate \omega_n 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.
• m y'' + k y = 0 gives simple harmonic motion at \omega_n.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k}{m}}$$

2. Step 2 — Rearrange symbolically for \omega_n:

   $$\omega_{n} = \omega_n = \sqrt{\dfrac{k}{m}}$$

3. Step 3

   $$List the givens: stiffness (k) = 21,625 N/m, mass (m) = 295.8 kg$$

4. Step 4 — Substitute the given values:

   $$\omega_{n} = \omega_n = \sqrt{\dfrac{21625}{295.8}}$$

5. Step 5 — Evaluate:

   $$\omega_{n} = 8.5503\ \text{rad/s}$$

6. Step 6 — Check: returning \omega_n = 8.5503 rad/s to

   $$\omega_n = \sqrt{\dfrac{k}{m}}$$

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

Why the other options are there

• 17.1005 — kept a factor of two that cancels in the correct rearrangement.
• 4.2751 — dropped that same factor in the other direction.
• 9.4053 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (Second Order)

A student verifies the solution of a linear differential equation at a given time. Given initial condition (y0) = 31.0000; rate constant (k) = 0.1000 1/s; time (t) = 0.9000 s, determine the solution value (y).

Start with the thinking

• The governing relation printed in this handbook section is First-order linear differential equation solution.
• 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.
• This first-order differential equation dy/dt = -k y has the exponential decay solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

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

2. Step 2 — Rearrange symbolically for y:

   $$y = y_0 e^{-kt}$$

3. Step 3 — List the givens: initial condition (y0) = 31.0000, rate constant (k) = 0.1000 1/s, time (t) = 0.9000 s.

4. Step 4 — Substitute the given values:

   $$y = y_0 e^{-k0.9000}$$

5. Step 5 — Evaluate:

   $$y = 28.3319$$

6. Step 6 — Check: returning y = 28.3319 to

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

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

Why the other options are there

• 56.6637 — kept a factor of two that cancels in the correct rearrangement.
• 14.1659 — dropped that same factor in the other direction.
• 31.1651 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations

A mathematics problem uses Slope–intercept line. Given x-value (x) = 5.5000; intercept (b) = 0.5000; y-value (y) = -6.8000, determine the slope (m).

Start with the thinking

• The governing relation printed in this handbook section is Slope–intercept line.
• Everything except m is given, so isolate m 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:

   $$y = m x + b$$

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

3. Step 3

   $$List the givens: x-value (x) = 5.5000, intercept (b) = 0.5000, y-value (y) = -6.8000$$

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

5. Step 5 — Evaluate:

   $$m = -1.3273$$

6. Step 6 — Check: returning m = -1.3273 to

   $$y = m x + b$$

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

Why the other options are there

• -2.6545 — kept a factor of two that cancels in the correct rearrangement.
• -0.6636 — dropped that same factor in the other direction.
• -1.4600 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Nonhomogeneous Differential Equations

A mathematics problem uses Exponential decay. Given decay constant (k) = 0.5000 1/s; time (t) = 0.4500 s; value at t (y) = 49.6900, 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

   $$List the givens: decay constant (k) = 0.5000 1/s, time (t) = 0.4500 s, value at t (y) = 49.6900$$

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

5. Step 5 — Evaluate:

   $$y_{0} = 62.2279$$

6. Step 6 — Check: returning y0 = 62.2279 to

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

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

Why the other options are there

• 124.5 — kept a factor of two that cancels in the correct rearrangement.
• 31.1140 — dropped that same factor in the other direction.
• 68.4507 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → First-Order Linear Nonhomogeneous Differential Equations

chlorine residual decay in a storage tank Given value at time t (y) = 4.8000 mg/L; rate constant (k) = 0.8000 1/h; elapsed time (t) = 1.6000 h, determine the initial value (y_0) in mg/L.

Start with the thinking

• The governing relation printed in this handbook section is First-order linear ODE solution.
• Everything except y_0 is given, so isolate y_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.
• The ODE y' + k y = 0 has the exponential solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

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

2. Step 2 — Rearrange symbolically for y_0:

   $$y_{0} = y_0 = y e^{k t}$$

3. Step 3 — List the givens: value at time t (y) = 4.8000 mg/L, rate constant (k) = 0.8000 1/h, elapsed time (t) = 1.6000 h.

4. Step 4 — Substitute the given values:

   $$y_{0} = y_0 = 4.8000 e^{0.8000 1.6000}$$

5. Step 5 — Evaluate:

   $$y_{0} = 17.2639\ \text{mg/L}$$

6. Step 6 — Check: returning y_0 = 17.2639 mg/L to

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

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

Why the other options are there

• 34.5277 — kept a factor of two that cancels in the correct rearrangement.
• 8.6319 — dropped that same factor in the other direction.
• 18.9903 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (First Order)

m y'' + k y = 0 gives simple harmonic motion at \omega_n. Given natural frequency (\omega_n) = 49.0000 rad/s; mass (m) = 261.5 kg, determine the stiffness (k) in N/m.

Start with the thinking

• The governing relation printed in this handbook section is Undamped natural frequency (2nd-order ODE).
• 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.
• m y'' + k y = 0 gives simple harmonic motion at \omega_n.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\omega_n = \sqrt{\dfrac{k}{m}}$$

2. Step 2 — Rearrange symbolically for k:

   $$k = k = m\,\omega_n^2$$

3. Step 3

   $$List the givens: natural frequency (\omega_n) = 49.0000 rad/s, mass (m) = 261.5 kg$$

4. Step 4 — Substitute the given values:

   $$k = k = 261.5\,\omega_n^2$$

5. Step 5 — Evaluate:

   $$k = 627862\ \text{N/m}$$

6. Step 6 — Check: returning k = 627,862 N/m to

   $$\omega_n = \sqrt{\dfrac{k}{m}}$$

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

Why the other options are there

• 1,255,723 — kept a factor of two that cancels in the correct rearrangement.
• 313,931 — dropped that same factor in the other direction.
• 690,648 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations (Second Order)

The differential equation for radioactive decay is solved for the initial quantity. Given rate constant (k) = 1.4000 1/s; time (t) = 3.3500 s; solution value (y) = 13.8100, determine the initial condition (y0).

Start with the thinking

• The governing relation printed in this handbook section is First-order linear differential equation solution.
• 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.
• This first-order differential equation dy/dt = -k y has the exponential decay solution shown.

Step-by-step solution

1. Step 1 — State the governing relation:

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

2. Step 2 — Rearrange symbolically for y0:

   $$y_{0} = \dfrac{y}{e^{-kt}}$$

3. Step 3 — List the givens: rate constant (k) = 1.4000 1/s, time (t) = 3.3500 s, solution value (y) = 13.8100.

4. Step 4 — Substitute the given values:

   $$y_{0} = \dfrac{13.8100}{e^{-k3.3500}}$$

5. Step 5 — Evaluate:

   $$y_{0} = 1503$$

6. Step 6 — Check: returning y0 = 1,503 to

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

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

Why the other options are there

• 3,007 — kept a factor of two that cancels in the correct rearrangement.
• 751.6 — dropped that same factor in the other direction.
• 1,654 — rounded an intermediate value before the final step.

Reference: FE Handbook — Differential Equations