A mathematics problem uses Exponential decay. Given initial value (y0) = 23.5000; decay constant (k) = 1.5500 1/s; time (t) = 2.5000 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) = 1.5500 1/s, time (t) = 2.5000 s.

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

5. Step 5 — Evaluate:

   $$y = 0.4877$$

6. Step 6 — Check: returning y = 0.4877 to

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

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

Why the other options are there

• 0.9755 — kept a factor of two that cancels in the correct rearrangement.
• 0.2439 — dropped that same factor in the other direction.
• 0.5365 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Dispersion, Mean, Median, and Mode Values

A student computes the mean of a data set to assess its dispersion. Given sum of values (sumx) = 476.0; number of values (n) = 30.0000; standard deviation (dispersion) (s) = 13.1300, determine the mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Dispersion, mean, median, and mode.
• Everything except xbar is given, so isolate xbar 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 mean is the central measure used with median and mode to describe the dispersion of a data set.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \dfrac{\sum x_i}{n}$$

3. Step 3 — List the givens: sum of values (sumx) = 476.0, number of values (n) = 30.0000, standard deviation (dispersion) (s) = 13.1300.

4. Step 4 — Substitute the given values:

   $$xbar = \dfrac{\sum x_i}{30.0000}$$

5. Step 5 — Evaluate:

   $$xbar = 15.8667$$

6. Step 6 — Check: returning xbar = 15.8667 to

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

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

Why the other options are there

• 31.7333 — kept a factor of two that cancels in the correct rearrangement.
• 7.9333 — dropped that same factor in the other direction.
• 17.4533 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dispersion, Mean, Median, and Mode Values

A mathematics problem uses Exponential decay. Given decay constant (k) = 1.4000 1/s; time (t) = 2.8500 s; value at t (y) = 32.6100, 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

   $$List the givens: decay constant (k) = 1.4000 1/s, time (t) = 2.8500 s, value at t (y) = 32.6100$$

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

5. Step 5 — Evaluate:

   $$y_{0} = 1763$$

6. Step 6 — Check: returning y0 = 1,763 to

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

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

Why the other options are there

• 3,525 — kept a factor of two that cancels in the correct rearrangement.
• 881.4 — dropped that same factor in the other direction.
• 1,939 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Dispersion, Mean, Median, and Mode Values

The mean value of test results is compared with the median and mode. Given number of values (n) = 24.0000; mean (xbar) = 94.1300; standard deviation (dispersion) (s) = 9.3800, determine the sum of values (sumx).

Start with the thinking

• The governing relation printed in this handbook section is Dispersion, mean, median, and mode.
• Everything except sumx is given, so isolate sumx 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 mean is the central measure used with median and mode to describe the dispersion of a data set.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

2. Step 2 — Rearrange symbolically for sumx:

   $$sumx = \bar{x}\, n$$

3. Step 3 — List the givens: number of values (n) = 24.0000, mean (xbar) = 94.1300, standard deviation (dispersion) (s) = 9.3800.

4. Step 4 — Substitute the given values:

   $$sumx = \bar{x}\, 24.0000$$

5. Step 5 — Evaluate:

   $$sumx = 2259$$

6. Step 6 — Check: returning sumx = 2,259 to

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

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

Why the other options are there

• 4,518 — kept a factor of two that cancels in the correct rearrangement.
• 1,130 — dropped that same factor in the other direction.
• 2,485 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dispersion, Mean, Median, and Mode Values

A mathematics problem uses Exponential decay. Given initial value (y0) = 13.0000; time (t) = 1.0000 s; value at t (y) = 26.6400, 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

   $$List the givens: initial value (y_{0}) = 13.0000, time (t) = 1.0000 s, value at t (y) = 26.6400$$

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

5. Step 5 — Evaluate:

   $$k = -0.7175\ \text{1/s}$$

6. Step 6 — Check: returning k = -0.7175 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

• -1.4349 — kept a factor of two that cancels in the correct rearrangement.
• -0.3587 — dropped that same factor in the other direction.
• -0.7892 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Dispersion, Mean, Median, and Mode Values

A quality engineer reports the mean, median, and mode of a batch of measurements. Given sum of values (sumx) = 221.0; mean (xbar) = 51.3200; standard deviation (dispersion) (s) = 18.8100, determine the number of values (n).

Start with the thinking

• The governing relation printed in this handbook section is Dispersion, mean, median, and mode.
• Everything except n is given, so isolate 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.
• The mean is the central measure used with median and mode to describe the dispersion of a data set.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

2. Step 2 — Rearrange symbolically for n:

   $$n = \dfrac{\sum x_i}{\bar{x}}$$

3. Step 3 — List the givens: sum of values (sumx) = 221.0, mean (xbar) = 51.3200, standard deviation (dispersion) (s) = 18.8100.

4. Step 4 — Substitute the given values:

   $$n = \dfrac{\sum x_i}{\bar{x}}$$

5. Step 5 — Evaluate:

   $$n = 4.3063$$

6. Step 6 — Check: returning n = 4.3063 to

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

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

Why the other options are there

• 8.6126 — kept a factor of two that cancels in the correct rearrangement.
• 2.1532 — dropped that same factor in the other direction.
• 4.7369 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dispersion, Mean, Median, and Mode Values

A mathematics problem uses Exponential decay. Given initial value (y0) = 47.0000; decay constant (k) = 0.6500 1/s; value at t (y) = 38.5100, 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) = 47.0000, decay constant (k) = 0.6500 1/s, value at t (y) = 38.5100.

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

5. Step 5 — Evaluate:

   $$t = 0.3065\ \text{s}$$

6. Step 6 — Check: returning t = 0.3065 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

• 0.6130 — kept a factor of two that cancels in the correct rearrangement.
• 0.1533 — dropped that same factor in the other direction.
• 0.3372 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Dispersion, Mean, Median, and Mode Values

A student computes the mean of a data set to assess its dispersion. Given sum of values (sumx) = 75.0000; number of values (n) = 23.0000; standard deviation (dispersion) (s) = 1.3600, determine the mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Dispersion, mean, median, and mode.
• Everything except xbar is given, so isolate xbar 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 mean is the central measure used with median and mode to describe the dispersion of a data set.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \dfrac{\sum x_i}{n}$$

3. Step 3 — List the givens: sum of values (sumx) = 75.0000, number of values (n) = 23.0000, standard deviation (dispersion) (s) = 1.3600.

4. Step 4 — Substitute the given values:

   $$xbar = \dfrac{\sum x_i}{23.0000}$$

5. Step 5 — Evaluate:

   $$xbar = 3.2609$$

6. Step 6 — Check: returning xbar = 3.2609 to

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

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

Why the other options are there

• 6.5217 — kept a factor of two that cancels in the correct rearrangement.
• 1.6304 — dropped that same factor in the other direction.
• 3.5870 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dispersion, Mean, Median, and Mode Values

A mathematics problem uses Exponential decay. Given initial value (y0) = 5.5000; decay constant (k) = 1.0000 1/s; time (t) = 1.6000 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) = 5.5000, decay constant (k) = 1.0000 1/s, time (t) = 1.6000 s.

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

5. Step 5 — Evaluate:

   $$y = 1.1104$$

6. Step 6 — Check: returning y = 1.1104 to

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

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

Why the other options are there

• 2.2209 — kept a factor of two that cancels in the correct rearrangement.
• 0.5552 — dropped that same factor in the other direction.
• 1.2215 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Dispersion, Mean, Median, and Mode Values

The mean value of test results is compared with the median and mode. Given number of values (n) = 14.0000; mean (xbar) = 75.9800; standard deviation (dispersion) (s) = 2.6800, determine the sum of values (sumx).

Start with the thinking

• The governing relation printed in this handbook section is Dispersion, mean, median, and mode.
• Everything except sumx is given, so isolate sumx 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 mean is the central measure used with median and mode to describe the dispersion of a data set.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

2. Step 2 — Rearrange symbolically for sumx:

   $$sumx = \bar{x}\, n$$

3. Step 3 — List the givens: number of values (n) = 14.0000, mean (xbar) = 75.9800, standard deviation (dispersion) (s) = 2.6800.

4. Step 4 — Substitute the given values:

   $$sumx = \bar{x}\, 14.0000$$

5. Step 5 — Evaluate:

   $$sumx = 1064$$

6. Step 6 — Check: returning sumx = 1,064 to

   $$\bar{x} = \dfrac{\sum x_i}{n}$$

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

Why the other options are there

• 2,127 — kept a factor of two that cancels in the correct rearrangement.
• 531.9 — dropped that same factor in the other direction.
• 1,170 — rounded an intermediate value before the final step.

Reference: FE Handbook — Dispersion, Mean, Median, and Mode Values