A mathematics problem uses Parabola vertex form. Given coefficient (a) = 2.0000; x-value (x) = 3.0000; vertex x (h) = 0.0000; vertex y (k) = -1.0000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Parabola vertex form.
• 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:

   $$y = a (x - h)^2 + k$$

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

3. Step 3

   $$List the givens: coefficient (a) = 2.0000, x-value (x) = 3.0000, vertex x (h) = 0.0000, vertex y (k) = -1.0000$$

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

5. Step 5 — Evaluate:

   $$y = 17.0000$$

6. Step 6 — Check: returning y = 17.0000 to

   $$y = a (x - h)^2 + k$$

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

Why the other options are there

• 34.0000 — kept a factor of two that cancels in the correct rearrangement.
• 8.5000 — dropped that same factor in the other direction.
• 18.7000 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Conic Section Equation

an elliptical culvert opening Given semi-major axis (a) = 2.3000 m; semi-minor axis (b) = 1.8000 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

   $$List the givens: semi-major axis (a) = 2.3000 m, semi-minor axis (b) = 1.8000 m$$

4. Step 4 — Substitute the given values:

   $$A = A = \pi 2.3000 1.8000$$

5. Step 5 — Evaluate:

   A = 13.0062\ \text{m^2}

6. Step 6 — Check: returning A = 13.0062 m^2 to

   $$A = \pi a b$$

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

Why the other options are there

• 26.0124 — kept a factor of two that cancels in the correct rearrangement.
• 6.5031 — dropped that same factor in the other direction.
• 14.3068 — 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) = 6.8000 m; semi-minor axis (b) = 4.4000 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

   $$List the givens: semi-major axis (a) = 6.8000 m, semi-minor axis (b) = 4.4000 m$$

4. Step 4 — Substitute the given values:

   $$e = e = \dfrac{\sqrt{6.8000^2 - 4.4000^2}}{6.8000}$$

5. Step 5 — Evaluate:

   $$e = 0.7624$$

6. Step 6 — Check: returning e = 0.7624 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

• 1.5249 — kept a factor of two that cancels in the correct rearrangement.
• 0.3812 — dropped that same factor in the other direction.
• 0.8387 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

A vertical curve profile is modeled as a parabola. Given curvature coefficient (a) = 0.1000 1/m; station offset (x) = 2.4000 m; vertex station (h) = 0.4000 m; vertex elevation (k) = 13.3000 m, determine the elevation (y) in m.

Start with the thinking

• The governing relation printed in this handbook section is Parabola (vertex form).
• 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.
• A vertical curve profile is modeled as a parabola.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a(x - h)^2 + k$$

2. Step 2 — Rearrange symbolically for y:

   $$y = y = a(x - h)^2 + k$$

3. Step 3 — List the givens: curvature coefficient (a) = 0.1000 1/m, station offset (x) = 2.4000 m, vertex station (h) = 0.4000 m, vertex elevation (k) = 13.3000 m.

4. Step 4 — Substitute the given values:

   $$y = y = 0.1000(2.4000 - 0.4000)^2 + 13.3000$$

5. Step 5 — Evaluate:

   $$y = 13.7000\ \text{m}$$

6. Step 6 — Check: returning y = 13.7000 m to

   $$y = a(x - h)^2 + k$$

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

Why the other options are there

• 27.4000 — kept a factor of two that cancels in the correct rearrangement.
• 6.8500 — dropped that same factor in the other direction.
• 15.0700 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Parabola)

A student evaluates a conic section equation at a given x-value. Given quadratic coefficient (a) = 2.9000; linear coefficient (b) = -0.5000; constant term (c) = 0.1000; x-value (x) = 0.9000, determine the y-value (y).

Start with the thinking

• The governing relation printed in this handbook section is Conic section equation — parabola form.
• 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 general conic section equation for a parabola relates its coefficients to a point on the curve.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a x^2 + b x + c$$

2. Step 2 — Rearrange symbolically for y:

   $$y = a x^2 + b x + c$$

3. Step 3 — List the givens: quadratic coefficient (a) = 2.9000, linear coefficient (b) = -0.5000, constant term (c) = 0.1000, x-value (x) = 0.9000.

4. Step 4 — Substitute the given values:

   $$y = 2.9000 0.9000^2 + -0.5000 0.9000 + 0.1000$$

5. Step 5 — Evaluate:

   $$y = 1.9990$$

6. Step 6 — Check: returning y = 1.9990 to

   $$y = a x^2 + b x + c$$

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

Why the other options are there

• 3.9980 — kept a factor of two that cancels in the correct rearrangement.
• 0.9995 — dropped that same factor in the other direction.
• 2.1989 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Section Equation

A mathematics problem uses Parabola vertex form. Given x-value (x) = 5.0000; vertex x (h) = 1.0000; vertex y (k) = 5.0000; y-value (y) = -36.0000, determine the coefficient (a).

Start with the thinking

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

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a (x - h)^2 + k$$

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

3. Step 3

   $$List the givens: x-value (x) = 5.0000, vertex x (h) = 1.0000, vertex y (k) = 5.0000, y-value (y) = -36.0000$$

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

5. Step 5 — Evaluate:

   $$a = -2.5625$$

6. Step 6 — Check: returning a = -2.5625 to

   $$y = a (x - h)^2 + k$$

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

Why the other options are there

• -5.1250 — kept a factor of two that cancels in the correct rearrangement.
• -1.2813 — dropped that same factor in the other direction.
• -2.8188 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Mathematics → Conic Section Equation

an elliptical bearing plate Given area (A) = 7.2000 m^2; semi-minor axis (b) = 1.5000 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

   $$List the givens: area (A) = 7.2000 m^2, semi-minor axis (b) = 1.5000 m$$

4. Step 4 — Substitute the given values:

   $$a = a = \dfrac{7.2000}{\pi 1.5000}$$

5. Step 5 — Evaluate:

   $$a = 1.5279\ \text{m}$$

6. Step 6 — Check: returning a = 1.5279 m to

   $$A = \pi a b$$

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

Why the other options are there

• 3.0558 — kept a factor of two that cancels in the correct rearrangement.
• 0.7639 — dropped that same factor in the other direction.
• 1.6807 — 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) = 0.8000; semi-major axis (a) = 2.6000 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

   $$List the givens: eccentricity (e) = 0.8000, semi-major axis (a) = 2.6000 m$$

4. Step 4 — Substitute the given values:

   $$b = b = 2.6000\sqrt{1 - 0.8000^2}$$

5. Step 5 — Evaluate:

   $$b = 1.5600\ \text{m}$$

6. Step 6 — Check: returning b = 1.5600 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

• 3.1200 — kept a factor of two that cancels in the correct rearrangement.
• 0.7800 — dropped that same factor in the other direction.
• 1.7160 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Ellipse)

A vertical curve profile is modeled as a parabola. Given elevation (y) = 11.2000 m; station offset (x) = 4.6000 m; vertex station (h) = 1.8000 m; vertex elevation (k) = 3.8000 m, determine the curvature coefficient (a) in 1/m.

Start with the thinking

• The governing relation printed in this handbook section is Parabola (vertex form).
• 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 vertical curve profile is modeled as a parabola.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a(x - h)^2 + k$$

2. Step 2 — Rearrange symbolically for a:

   $$a = a = \dfrac{y - k}{(x - h)^2}$$

3. Step 3 — List the givens: elevation (y) = 11.2000 m, station offset (x) = 4.6000 m, vertex station (h) = 1.8000 m, vertex elevation (k) = 3.8000 m.

4. Step 4 — Substitute the given values:

   $$a = a = \dfrac{11.2000 - 3.8000}{(4.6000 - 1.8000)^2}$$

5. Step 5 — Evaluate:

   $$a = 0.9439\ \text{1/m}$$

6. Step 6 — Check: returning a = 0.9439 1/m to

   $$y = a(x - h)^2 + k$$

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

Why the other options are there

• 1.8878 — kept a factor of two that cancels in the correct rearrangement.
• 0.4719 — dropped that same factor in the other direction.
• 1.0383 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Sections (Parabola)

A designer fits a conic section equation to a vertical curve. Given quadratic coefficient (a) = 3.0000; linear coefficient (b) = -3.4000; x-value (x) = -1.9000; y-value (y) = -10.1000, determine the constant term (c).

Start with the thinking

• The governing relation printed in this handbook section is Conic section equation — parabola form.
• Everything except c is given, so isolate c 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 general conic section equation for a parabola relates its coefficients to a point on the curve.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$y = a x^2 + b x + c$$

2. Step 2 — Rearrange symbolically for c:

   $$c = y - a x^2 - b x$$

3. Step 3 — List the givens: quadratic coefficient (a) = 3.0000, linear coefficient (b) = -3.4000, x-value (x) = -1.9000, y-value (y) = -10.1000.

4. Step 4 — Substitute the given values:

   $$c = -10.1000 - 3.0000 -1.9000^2 - -3.4000 -1.9000$$

5. Step 5 — Evaluate:

   $$c = -27.3900$$

6. Step 6 — Check: returning c = -27.3900 to

   $$y = a x^2 + b x + c$$

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

Why the other options are there

• -54.7800 — kept a factor of two that cancels in the correct rearrangement.
• -13.6950 — dropped that same factor in the other direction.
• -30.1290 — rounded an intermediate value before the final step.

Reference: FE Handbook — Conic Section Equation