an elliptical manhole cover Given semi-major axis (a) = 5.9000 m; semi-minor axis (b) = 2.0000 m, determine the area (A) in m^2.

Start with the thinking

• The governing relation printed in this handbook section is Area of an ellipse.
• 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.
• A survey monument plate is machined as an ellipse.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A = \pi a b$$

2. Step 2 — Rearrange symbolically for A:

   $$A = A = \pi a b$$

3. Step 3

   $$List the givens: semi-major axis (a) = 5.9000 m, semi-minor axis (b) = 2.0000 m$$

4. Step 4 — Substitute the given values:

   $$A = A = \pi 5.9000 2.0000$$

5. Step 5 — Evaluate:

   A = 37.0708\ \text{m^2}

6. Step 6 — Check: returning A = 37.0708 m^2 to

   $$A = \pi a b$$

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

Why the other options are there

• 74.1416 — kept a factor of two that cancels in the correct rearrangement.
• 18.5354 — dropped that same factor in the other direction.
• 40.7779 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

Eccentricity measures how far the conic departs from a circle. Given semi-major axis (a) = 3.9000 m; semi-minor axis (b) = 2.6000 m, determine the eccentricity (e).

Start with the thinking

• The governing relation printed in this handbook section is Eccentricity of an ellipse.
• Everything except e is given, so isolate e 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.
• Eccentricity measures how far the conic departs from a circle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

2. Step 2 — Rearrange symbolically for e:

   $$e = e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

3. Step 3

   $$List the givens: semi-major axis (a) = 3.9000 m, semi-minor axis (b) = 2.6000 m$$

4. Step 4 — Substitute the given values:

   $$e = e = \dfrac{\sqrt{3.9000^2 - 2.6000^2}}{3.9000}$$

5. Step 5 — Evaluate:

   $$e = 0.7454$$

6. Step 6 — Check: returning e = 0.7454 to

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

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

Why the other options are there

• 1.4907 — kept a factor of two that cancels in the correct rearrangement.
• 0.3727 — dropped that same factor in the other direction.
• 0.8199 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

A student computes the eccentricity of an ellipse from its axes. Given semi-major axis (a) = 19.5000 ft; semi-minor axis (b) = 6.0000 ft, determine the eccentricity (e).

Start with the thinking

• The governing relation printed in this handbook section is Ellipse — semi-axes and eccentricity.
• Everything except e is given, so isolate e 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 ellipse relation ties the semi-major axis, semi-minor axis, and eccentricity of the conic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

2. Step 2 — Rearrange symbolically for e:

   $$e = \dfrac{\sqrt{a^2-b^2}}{a}$$

3. Step 3

   $$List the givens: semi-major axis (a) = 19.5000 ft, semi-minor axis (b) = 6.0000 ft$$

4. Step 4 — Substitute the given values:

   $$e = \dfrac{\sqrt{19.5000^2-6.0000^2}}{19.5000}$$

5. Step 5 — Evaluate:

   $$e = 0.9515$$

6. Step 6 — Check: returning e = 0.9515 to

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

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

Why the other options are there

• 1.9030 — kept a factor of two that cancels in the correct rearrangement.
• 0.4757 — dropped that same factor in the other direction.
• 1.0466 — rounded an intermediate value before the final step.

Reference: FE Handbook — Ellipse

an elliptical culvert opening Given area (A) = 25.1000 m^2; semi-minor axis (b) = 2.3000 m, determine the semi-major axis (a) in m.

Start with the thinking

• The governing relation printed in this handbook section is Area of an ellipse.
• 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.
• A survey monument plate is machined as an ellipse.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A = \pi a b$$

2. Step 2 — Rearrange symbolically for a:

   $$a = a = \dfrac{A}{\pi b}$$

3. Step 3

   $$List the givens: area (A) = 25.1000 m^2, semi-minor axis (b) = 2.3000 m$$

4. Step 4 — Substitute the given values:

   $$a = a = \dfrac{25.1000}{\pi 2.3000}$$

5. Step 5 — Evaluate:

   $$a = 3.4737\ \text{m}$$

6. Step 6 — Check: returning a = 3.4737 m to

   $$A = \pi a b$$

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

Why the other options are there

• 6.9475 — kept a factor of two that cancels in the correct rearrangement.
• 1.7369 — dropped that same factor in the other direction.
• 3.8211 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

Eccentricity measures how far the conic departs from a circle. Given eccentricity (e) = 0.9000; semi-major axis (a) = 2.7000 m, determine the semi-minor axis (b) in m.

Start with the thinking

• The governing relation printed in this handbook section is Eccentricity of an ellipse.
• Everything except b is given, so isolate b 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.
• Eccentricity measures how far the conic departs from a circle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = b = a\sqrt{1 - e^2}$$

3. Step 3

   $$List the givens: eccentricity (e) = 0.9000, semi-major axis (a) = 2.7000 m$$

4. Step 4 — Substitute the given values:

   $$b = b = 2.7000\sqrt{1 - 0.9000^2}$$

5. Step 5 — Evaluate:

   $$b = 1.1769\ \text{m}$$

6. Step 6 — Check: returning b = 1.1769 m to

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

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

Why the other options are there

• 2.3538 — kept a factor of two that cancels in the correct rearrangement.
• 0.5885 — dropped that same factor in the other direction.
• 1.2946 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

An engineer designs an elliptical arch and checks its eccentricity. Given semi-major axis (a) = 19.0000 ft; eccentricity (e) = 0.9130, determine the semi-minor axis (b) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Ellipse — semi-axes and eccentricity.
• Everything except b is given, so isolate b 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 ellipse relation ties the semi-major axis, semi-minor axis, and eccentricity of the conic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = a\sqrt{1-e^2}$$

3. Step 3

   $$List the givens: semi-major axis (a) = 19.0000 ft, eccentricity (e) = 0.9130$$

4. Step 4 — Substitute the given values:

   $$b = 19.0000\sqrt{1-0.9130^2}$$

5. Step 5 — Evaluate:

   $$b = 7.7512\ \text{ft}$$

6. Step 6 — Check: returning b = 7.7512 ft to

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

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

Why the other options are there

• 15.5025 — kept a factor of two that cancels in the correct rearrangement.
• 3.8756 — dropped that same factor in the other direction.
• 8.5264 — rounded an intermediate value before the final step.

Reference: FE Handbook — Ellipse

an elliptical bearing plate Given area (A) = 31.9000 m^2; semi-major axis (a) = 3.2000 m, determine the semi-minor axis (b) in m.

Start with the thinking

• The governing relation printed in this handbook section is Area of an ellipse.
• Everything except b is given, so isolate b 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.
• A survey monument plate is machined as an ellipse.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A = \pi a b$$

2. Step 2 — Rearrange symbolically for b:

   $$b = b = \dfrac{A}{\pi a}$$

3. Step 3

   $$List the givens: area (A) = 31.9000 m^2, semi-major axis (a) = 3.2000 m$$

4. Step 4 — Substitute the given values:

   $$b = b = \dfrac{31.9000}{\pi 3.2000}$$

5. Step 5 — Evaluate:

   $$b = 3.1732\ \text{m}$$

6. Step 6 — Check: returning b = 3.1732 m to

   $$A = \pi a b$$

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

Why the other options are there

• 6.3463 — kept a factor of two that cancels in the correct rearrangement.
• 1.5866 — dropped that same factor in the other direction.
• 3.4905 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

Eccentricity measures how far the conic departs from a circle. Given semi-major axis (a) = 5.2000 m; semi-minor axis (b) = 1.0000 m, determine the eccentricity (e).

Start with the thinking

• The governing relation printed in this handbook section is Eccentricity of an ellipse.
• Everything except e is given, so isolate e 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.
• Eccentricity measures how far the conic departs from a circle.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

2. Step 2 — Rearrange symbolically for e:

   $$e = e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

3. Step 3

   $$List the givens: semi-major axis (a) = 5.2000 m, semi-minor axis (b) = 1.0000 m$$

4. Step 4 — Substitute the given values:

   $$e = e = \dfrac{\sqrt{5.2000^2 - 1.0000^2}}{5.2000}$$

5. Step 5 — Evaluate:

   $$e = 0.9813$$

6. Step 6 — Check: returning e = 0.9813 to

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

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

Why the other options are there

• 1.9627 — kept a factor of two that cancels in the correct rearrangement.
• 0.4907 — dropped that same factor in the other direction.
• 1.0795 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

A designer lays out an ellipse for a decorative footing outline. Given semi-minor axis (b) = 4.5000 ft; eccentricity (e) = 0.6020, determine the semi-major axis (a) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Ellipse — semi-axes and eccentricity.
• 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 ellipse relation ties the semi-major axis, semi-minor axis, and eccentricity of the conic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{b}{\sqrt{1-e^2}}$$

3. Step 3

   $$List the givens: semi-minor axis (b) = 4.5000 ft, eccentricity (e) = 0.6020$$

4. Step 4 — Substitute the given values:

   $$a = \dfrac{4.5000}{\sqrt{1-0.6020^2}}$$

5. Step 5 — Evaluate:

   $$a = 5.6356\ \text{ft}$$

6. Step 6 — Check: returning a = 5.6356 ft to

   $$e = \dfrac{\sqrt{a^2 - b^2}}{a}$$

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

Why the other options are there

• 11.2712 — kept a factor of two that cancels in the correct rearrangement.
• 2.8178 — dropped that same factor in the other direction.
• 6.1992 — rounded an intermediate value before the final step.

Reference: FE Handbook — Ellipse

an elliptical manhole cover Given semi-major axis (a) = 1.3000 m; semi-minor axis (b) = 2.2000 m, determine the area (A) in m^2.

Start with the thinking

• The governing relation printed in this handbook section is Area of an ellipse.
• 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.
• A survey monument plate is machined as an ellipse.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$A = \pi a b$$

2. Step 2 — Rearrange symbolically for A:

   $$A = A = \pi a b$$

3. Step 3

   $$List the givens: semi-major axis (a) = 1.3000 m, semi-minor axis (b) = 2.2000 m$$

4. Step 4 — Substitute the given values:

   $$A = A = \pi 1.3000 2.2000$$

5. Step 5 — Evaluate:

   A = 8.9850\ \text{m^2}

6. Step 6 — Check: returning A = 8.9850 m^2 to

   $$A = \pi a b$$

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

Why the other options are there

• 17.9699 — kept a factor of two that cancels in the correct rearrangement.
• 4.4925 — dropped that same factor in the other direction.
• 9.8835 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)