A tank designer sizes a spherical storage vessel. Given radius (r) = 8.8000 ft; surface area (S) = 697.0 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

   $$List the givens: radius (r) = 8.8000 ft, surface area (S) = 697.0 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 8.8000^3$$

5. Step 5 — Evaluate:

   V = 2855\ \text{ft^3}

6. Step 6 — Check: returning V = 2,855 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

• 5,709 — kept a factor of two that cancels in the correct rearrangement.
• 1,427 — dropped that same factor in the other direction.
• 3,140 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

An engineer computes the volume of a spherical dome. Given radius (r) = 7.9000 ft; volume (V) = 46.0000 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

   $$List the givens: radius (r) = 7.9000 ft, volume (V) = 46.0000 ft^3$$

4. Step 4 — Substitute the given values:

   $$S = 4\pi 7.9000^2$$

5. Step 5 — Evaluate:

   S = 784.3\ \text{ft^2}

6. Step 6 — Check: returning S = 784.3 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

• 1,569 — kept a factor of two that cancels in the correct rearrangement.
• 392.1 — dropped that same factor in the other direction.
• 862.7 — 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) = 4,045 ft^3; surface area (S) = 812.0 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

   $$List the givens: volume (V) = 4,045 ft^3, surface area (S) = 812.0 ft^2$$

4. Step 4 — Substitute the given values:

   $$r = \sqrt[3]{\dfrac{34045}{4\pi}}$$

5. Step 5 — Evaluate:

   $$r = 9.8838\ \text{ft}$$

6. Step 6 — Check: returning r = 9.8838 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

• 19.7677 — kept a factor of two that cancels in the correct rearrangement.
• 4.9419 — dropped that same factor in the other direction.
• 10.8722 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A tank designer sizes a spherical storage vessel. Given radius (r) = 5.5000 ft; surface area (S) = 1,044 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

   $$List the givens: radius (r) = 5.5000 ft, surface area (S) = 1,044 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 5.5000^3$$

5. Step 5 — Evaluate:

   V = 696.9\ \text{ft^3}

6. Step 6 — Check: returning V = 696.9 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

• 1,394 — kept a factor of two that cancels in the correct rearrangement.
• 348.5 — dropped that same factor in the other direction.
• 766.6 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

An engineer computes the volume of a spherical dome. Given radius (r) = 4.0000 ft; volume (V) = 1,611 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

   $$List the givens: radius (r) = 4.0000 ft, volume (V) = 1,611 ft^3$$

4. Step 4 — Substitute the given values:

   $$S = 4\pi 4.0000^2$$

5. Step 5 — Evaluate:

   S = 201.1\ \text{ft^2}

6. Step 6 — Check: returning S = 201.1 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

• 402.1 — kept a factor of two that cancels in the correct rearrangement.
• 100.5 — dropped that same factor in the other direction.
• 221.2 — 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) = 2,412 ft^3; surface area (S) = 1,227 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

   $$List the givens: volume (V) = 2,412 ft^3, surface area (S) = 1,227 ft^2$$

4. Step 4 — Substitute the given values:

   $$r = \sqrt[3]{\dfrac{32412}{4\pi}}$$

5. Step 5 — Evaluate:

   $$r = 8.3196\ \text{ft}$$

6. Step 6 — Check: returning r = 8.3196 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

• 16.6392 — kept a factor of two that cancels in the correct rearrangement.
• 4.1598 — dropped that same factor in the other direction.
• 9.1516 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A tank designer sizes a spherical storage vessel. Given radius (r) = 2.5000 ft; surface area (S) = 104.5 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

   $$List the givens: radius (r) = 2.5000 ft, surface area (S) = 104.5 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 2.5000^3$$

5. Step 5 — Evaluate:

   V = 65.4498\ \text{ft^3}

6. Step 6 — Check: returning V = 65.4498 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

• 130.9 — kept a factor of two that cancels in the correct rearrangement.
• 32.7249 — dropped that same factor in the other direction.
• 71.9948 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

An engineer computes the volume of a spherical dome. Given radius (r) = 5.9000 ft; volume (V) = 490.5 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

   $$List the givens: radius (r) = 5.9000 ft, volume (V) = 490.5 ft^3$$

4. Step 4 — Substitute the given values:

   $$S = 4\pi 5.9000^2$$

5. Step 5 — Evaluate:

   S = 437.4\ \text{ft^2}

6. Step 6 — Check: returning S = 437.4 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

• 874.9 — kept a factor of two that cancels in the correct rearrangement.
• 218.7 — dropped that same factor in the other direction.
• 481.2 — 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) = 120.1 ft^3; surface area (S) = 410.1 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

   $$List the givens: volume (V) = 120.1 ft^3, surface area (S) = 410.1 ft^2$$

4. Step 4 — Substitute the given values:

   $$r = \sqrt[3]{\dfrac{3120.1}{4\pi}}$$

5. Step 5 — Evaluate:

   $$r = 3.0607\ \text{ft}$$

6. Step 6 — Check: returning r = 3.0607 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

• 6.1214 — kept a factor of two that cancels in the correct rearrangement.
• 1.5303 — dropped that same factor in the other direction.
• 3.3667 — rounded an intermediate value before the final step.

Reference: FE Handbook — Sphere

A tank designer sizes a spherical storage vessel. Given radius (r) = 4.1000 ft; surface area (S) = 340.1 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

   $$List the givens: radius (r) = 4.1000 ft, surface area (S) = 340.1 ft^2$$

4. Step 4 — Substitute the given values:

   $$V = \dfrac{4}{3}\pi 4.1000^3$$

5. Step 5 — Evaluate:

   V = 288.7\ \text{ft^3}

6. Step 6 — Check: returning V = 288.7 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

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

Reference: FE Handbook — Sphere