A cylindrical tank is 18 ft in diameter and 6 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Start with the thinking

• Volume uses the circular area times height.
• Total surface adds both end caps to the lateral shell.

Step-by-step solution

1. Volume

   $$V = (\pi/4)d^{2}h$$

2. Substituting

   $$V = (\pi/4)(18)^{2}(6) = 1,527 ft^{3}$$

3. Convert

   $$V = 1,527 \times 7.48 = 11,421 gal$$

4. Surface — S = πdh + 2(π/4)d²

5. Substituting

   $$S = \pi(18)(6) + 2(\pi/4)(18)^{2} = 848.2 ft^{2}$$

Why the other options are there

• 6,107 ft³ (radius and diameter confused)
• 339.3 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A circular sedimentation basin of radius 5.5 ft is partitioned by a central angle of 157°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 5.5 \times 2.7402 = 15.07 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(5.5)^{2}(2.7402) = 41.45 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(5.5)^{2}(2.7402 - 0.3907) = 35.54 ft^{2}$$

Why the other options are there

• 2,375 ft² (degrees used directly)
• 26.32 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A cylindrical tank is 15 ft in diameter and 28 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Start with the thinking

• Volume uses the circular area times height.
• Total surface adds both end caps to the lateral shell.

Step-by-step solution

1. Volume

   $$V = (\pi/4)d^{2}h$$

2. Substituting

   $$V = (\pi/4)(15)^{2}(28) = 4,948 ft^{3}$$

3. Convert

   $$V = 4,948 \times 7.48 = 37,011 gal$$

4. Surface — S = πdh + 2(π/4)d²

5. Substituting

   $$S = \pi(15)(28) + 2(\pi/4)(15)^{2} = 1,673 ft^{2}$$

Why the other options are there

• 19,792 ft³ (radius and diameter confused)
• 1,319 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A circular sedimentation basin of radius 6.0 ft is partitioned by a central angle of 109°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 6.0 \times 1.9024 = 11.41 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(6.0)^{2}(1.9024) = 34.24 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(6.0)^{2}(1.9024 - 0.9455) = 17.22 ft^{2}$$

Why the other options are there

• 1,962 ft² (degrees used directly)
• 16.24 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A cylindrical tank is 8 ft in diameter and 24 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Start with the thinking

• Volume uses the circular area times height.
• Total surface adds both end caps to the lateral shell.

Step-by-step solution

1. Volume

   $$V = (\pi/4)d^{2}h$$

2. Substituting

   $$V = (\pi/4)(8)^{2}(24) = 1,206 ft^{3}$$

3. Convert

   $$V = 1,206 \times 7.48 = 9,024 gal$$

4. Surface — S = πdh + 2(π/4)d²

5. Substituting

   $$S = \pi(8)(24) + 2(\pi/4)(8)^{2} = 703.7 ft^{2}$$

Why the other options are there

• 4,825 ft³ (radius and diameter confused)
• 603.2 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A circular sedimentation basin of radius 4.5 ft is partitioned by a central angle of 62°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 4.5 \times 1.0821 = 4.87 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(4.5)^{2}(1.0821) = 10.96 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(4.5)^{2}(1.0821 - 0.8829) = 2.02 ft^{2}$$

Why the other options are there

• 627.8 ft² (degrees used directly)
• 0.83 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A cylindrical tank is 14 ft in diameter and 12 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Start with the thinking

• Volume uses the circular area times height.
• Total surface adds both end caps to the lateral shell.

Step-by-step solution

1. Volume

   $$V = (\pi/4)d^{2}h$$

2. Substituting

   $$V = (\pi/4)(14)^{2}(12) = 1,847 ft^{3}$$

3. Convert

   $$V = 1,847 \times 7.48 = 13,817 gal$$

4. Surface — S = πdh + 2(π/4)d²

5. Substituting

   $$S = \pi(14)(12) + 2(\pi/4)(14)^{2} = 835.7 ft^{2}$$

Why the other options are there

• 7,389 ft³ (radius and diameter confused)
• 527.8 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A circular sedimentation basin of radius 7.5 ft is partitioned by a central angle of 110°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 7.5 \times 1.9199 = 14.40 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(7.5)^{2}(1.9199) = 54.00 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(7.5)^{2}(1.9199 - 0.9397) = 27.57 ft^{2}$$

Why the other options are there

• 3,094 ft² (degrees used directly)
• 25.87 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A cylindrical tank is 9 ft in diameter and 16 ft tall. Most nearly, what is its volume in gallons and its total surface area?

Start with the thinking

• Volume uses the circular area times height.
• Total surface adds both end caps to the lateral shell.

Step-by-step solution

1. Volume

   $$V = (\pi/4)d^{2}h$$

2. Substituting

   $$V = (\pi/4)(9)^{2}(16) = 1,018 ft^{3}$$

3. Convert

   $$V = 1,018 \times 7.48 = 7,614 gal$$

4. Surface — S = πdh + 2(π/4)d²

5. Substituting

   $$S = \pi(9)(16) + 2(\pi/4)(9)^{2} = 579.6 ft^{2}$$

Why the other options are there

• 4,072 ft³ (radius and diameter confused)
• 452.4 ft² (end caps omitted)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder

A circular sedimentation basin of radius 11.0 ft is partitioned by a central angle of 129°. Compute the arc length, the area of the circular sector, and the area of the circular segment cut off by the chord.

Start with the thinking

• All mensuration formulas for sectors and segments require the central angle in radians.
• The segment is the sector minus the triangle formed by the two radii and the chord.

Step-by-step solution

1. Formula

   $$s = R \theta$$

2. Substituting

   $$s = 11.0 \times 2.2515 = 24.77 ft$$

3. Formula

   $$A_sector = ½ R^{2} \theta$$

4. Substituting

   $$A_sector = ½(11.0)^{2}(2.2515) = 136.2 ft^{2}$$

5. Formula

   $$A_segment = ½ R^{2} (\theta - \sin \theta)$$

6. Substituting

   $$A_segment = ½(11.0)^{2}(2.2515 - 0.7771) = 89.20 ft^{2}$$

Why the other options are there

• 7,805 ft² (degrees used directly)
• 75.71 ft² (wrong triangle area)

Reference: FE Reference Handbook — Mathematics → Right Circular Cylinder