A tank designer sizes a spherical storage vessel. Given radius (r) = 7.5000 ft; surface area (S) = 676.4 ft^2, determine the volume (V) in ft^3.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except V is given, so isolate V 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for V:

   $$V = \dfrac{4}{3}\pi r^3$$

3. Step 3

   $$List the givens: radius (r) = 7.5000 ft, surface area (S) = 676.4 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 7.5000^3$$

5. Step 5 — Evaluate:

   V = 1767\ \text{ft^3}

6. Step 6 — Check: returning V = 1,767 ft^3 to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 3,534 — kept a factor of two that cancels in the correct rearrangement.
• 883.6 — dropped that same factor in the other direction.
• 1,944 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

An engineer computes the volume of a spherical dome. Given radius (r) = 1.7000 ft; volume (V) = 2,044 ft^3, determine the surface area (S) in ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except S is given, so isolate S 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for S:

   $$S = 4\pi r^2$$

3. Step 3

   $$List the givens: radius (r) = 1.7000 ft, volume (V) = 2,044 ft^3$$

4. Step 4 — Substitute the given values:

   $$S = 4\pi 1.7000^2$$

5. Step 5 — Evaluate:

   S = 36.3168\ \text{ft^2}

6. Step 6 — Check: returning S = 36.3168 ft^2 to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 72.6336 — kept a factor of two that cancels in the correct rearrangement.
• 18.1584 — dropped that same factor in the other direction.
• 39.9485 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A student finds the surface area of a sphere given its radius. Given volume (V) = 2,610 ft^3; surface area (S) = 287.1 ft^2, determine the radius (r) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except r is given, so isolate r 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for r:

   $$r = \sqrt[3]{\dfrac{3V}{4\pi}}$$

3. Step 3

   $$List the givens: volume (V) = 2,610 ft^3, surface area (S) = 287.1 ft^2$$

4. Step 4 — Substitute the given values:

   $$r = \sqrt[3]{\dfrac{32610}{4\pi}}$$

5. Step 5 — Evaluate:

   $$r = 8.5415\ \text{ft}$$

6. Step 6 — Check: returning r = 8.5415 ft to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 17.0830 — kept a factor of two that cancels in the correct rearrangement.
• 4.2707 — dropped that same factor in the other direction.
• 9.3956 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A tank designer sizes a spherical storage vessel. Given radius (r) = 5.3000 ft; surface area (S) = 259.1 ft^2, determine the volume (V) in ft^3.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except V is given, so isolate V 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for V:

   $$V = \dfrac{4}{3}\pi r^3$$

3. Step 3

   $$List the givens: radius (r) = 5.3000 ft, surface area (S) = 259.1 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 5.3000^3$$

5. Step 5 — Evaluate:

   V = 623.6\ \text{ft^3}

6. Step 6 — Check: returning V = 623.6 ft^3 to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 1,247 — kept a factor of two that cancels in the correct rearrangement.
• 311.8 — dropped that same factor in the other direction.
• 686.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

An engineer computes the volume of a spherical dome. Given radius (r) = 1.8000 ft; volume (V) = 3,631 ft^3, determine the surface area (S) in ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except S is given, so isolate S 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for S:

   $$S = 4\pi r^2$$

3. Step 3

   $$List the givens: radius (r) = 1.8000 ft, volume (V) = 3,631 ft^3$$

4. Step 4 — Substitute the given values:

   $$S = 4\pi 1.8000^2$$

5. Step 5 — Evaluate:

   S = 40.7150\ \text{ft^2}

6. Step 6 — Check: returning S = 40.7150 ft^2 to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 81.4301 — kept a factor of two that cancels in the correct rearrangement.
• 20.3575 — dropped that same factor in the other direction.
• 44.7865 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A student finds the surface area of a sphere given its radius. Given volume (V) = 1,970 ft^3; surface area (S) = 745.5 ft^2, determine the radius (r) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except r is given, so isolate r 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for r:

   $$r = \sqrt[3]{\dfrac{3V}{4\pi}}$$

3. Step 3

   $$List the givens: volume (V) = 1,970 ft^3, surface area (S) = 745.5 ft^2$$

4. Step 4 — Substitute the given values:

   $$r = \sqrt[3]{\dfrac{31970}{4\pi}}$$

5. Step 5 — Evaluate:

   $$r = 7.7769\ \text{ft}$$

6. Step 6 — Check: returning r = 7.7769 ft to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 15.5538 — kept a factor of two that cancels in the correct rearrangement.
• 3.8885 — dropped that same factor in the other direction.
• 8.5546 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A tank designer sizes a spherical storage vessel. Given radius (r) = 6.9000 ft; surface area (S) = 712.0 ft^2, determine the volume (V) in ft^3.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except V is given, so isolate V 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for V:

   $$V = \dfrac{4}{3}\pi r^3$$

3. Step 3

   $$List the givens: radius (r) = 6.9000 ft, surface area (S) = 712.0 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 6.9000^3$$

5. Step 5 — Evaluate:

   V = 1376\ \text{ft^3}

6. Step 6 — Check: returning V = 1,376 ft^3 to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 2,752 — kept a factor of two that cancels in the correct rearrangement.
• 688.0 — dropped that same factor in the other direction.
• 1,514 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

An engineer computes the volume of a spherical dome. Given radius (r) = 3.1000 ft; volume (V) = 1,367 ft^3, determine the surface area (S) in ft^2.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except S is given, so isolate S 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for S:

   $$S = 4\pi r^2$$

3. Step 3

   $$List the givens: radius (r) = 3.1000 ft, volume (V) = 1,367 ft^3$$

4. Step 4 — Substitute the given values:

   $$S = 4\pi 3.1000^2$$

5. Step 5 — Evaluate:

   S = 120.8\ \text{ft^2}

6. Step 6 — Check: returning S = 120.8 ft^2 to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 241.5 — kept a factor of two that cancels in the correct rearrangement.
• 60.3814 — dropped that same factor in the other direction.
• 132.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A student finds the surface area of a sphere given its radius. Given volume (V) = 3,467 ft^3; surface area (S) = 497.7 ft^2, determine the radius (r) in ft.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except r is given, so isolate r 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for r:

   $$r = \sqrt[3]{\dfrac{3V}{4\pi}}$$

3. Step 3

   $$List the givens: volume (V) = 3,467 ft^3, surface area (S) = 497.7 ft^2$$

4. Step 4 — Substitute the given values:

   $$r = \sqrt[3]{\dfrac{33467}{4\pi}}$$

5. Step 5 — Evaluate:

   $$r = 9.3894\ \text{ft}$$

6. Step 6 — Check: returning r = 9.3894 ft to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 18.7788 — kept a factor of two that cancels in the correct rearrangement.
• 4.6947 — dropped that same factor in the other direction.
• 10.3284 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A tank designer sizes a spherical storage vessel. Given radius (r) = 1.7000 ft; surface area (S) = 439.6 ft^2, determine the volume (V) in ft^3.

Start with the thinking

• The governing relation printed in this handbook section is Sphere volume and surface area.
• Everything except V is given, so isolate V 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 sphere volume and surface area formulas relate the radius of a sphere to its volume and area.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$V = \dfrac{4}{3}\pi r^3$$

2. Step 2 — Rearrange symbolically for V:

   $$V = \dfrac{4}{3}\pi r^3$$

3. Step 3

   $$List the givens: radius (r) = 1.7000 ft, surface area (S) = 439.6 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 1.7000^3$$

5. Step 5 — Evaluate:

   V = 20.5795\ \text{ft^3}

6. Step 6 — Check: returning V = 20.5795 ft^3 to

   $$V = \dfrac{4}{3}\pi r^3$$

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

Why the other options are there

• 41.1591 — kept a factor of two that cancels in the correct rearrangement.
• 10.2898 — dropped that same factor in the other direction.
• 22.6375 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere