3D Volume and Surface

lessonbunny.com

Answer Key & Explanations

3D Volume and Surface · Grade 7 · Worksheet 2

1. Liam is designing a cylindrical water tank for his school's science project. The tank needs to hold exactly 50,000 cubic centimeters of water. If the height of the tank is 80 centimeters, what must be the radius of the circular base? Use π ≈ 3.14 and round your answer to the nearest tenth of a centimeter. Answer: 14.1 Solution: We are given the volume of a cylindrical tank: 50,000 cubic centimeters. The height h = 80 cm. We need the radius r of the base.
   Full step-by-step solution

   We are given the volume of a cylindrical tank: 50,000 cubic centimeters. The height h = 80 cm. We need the radius r of the base. --- **Step 1: Recall the volume formula for a cylinder** Volume V = π × r² × h --- **Step 2: Substitute known values into the formula** 50,000 = 3.14 × r² × 80 --- **Step 3: Multiply 3.14 and 80** 3.14 × 80 = 251.2 So: 50,000 = 251.2 × r² --- **Step 4: Solve for r²** Divide both sides by 251.2: r² = 50,000 / 251.2 --- **Step 5: Perform the division** 50,000 ÷ 251.2 First, multiply numerator and denominator by 10 to avoid decimals: 500,000 ÷ 2512 Now divide: 2512 × 199 = 2512 × 200 − 2512 = 502,400 − 2512 = 499,888 500,000 − 499,888 = 112 So 500,000 ÷ 2512 ≈ 199 + (112 / 2512) 112 / 2512 ≈ 0.0446 Thus: 500,000 / 2512 ≈ 199.0446 But this is for 500,000 / 2512, which equals 50,000 / 251.2. So: 50,000 / 251.2 ≈ 199.0446 So r² ≈ 199.0446 --- **Step 6: Find r by taking the square root** r ≈ √199.0446 We know 14² = 196, 15² = 225. 14.1² = 198.81 14.2² = 201.64 199.0446 is slightly above 198.81, so r is slightly above 14.1. --- **Step 7: Refine the estimate** Difference from 198.81: 199.0446 − 198.81 = 0.2346 Increment from 14.1: slope ≈ 2 × 14.1 = 28.2 per unit, so per 0.01 in r, change ≈ 0.282 in r². We need 0.2346 increase: 0.2346 / 0.282 ≈ 0.832 × 0.01 ≈ 0.00832 So r ≈ 14.1083 --- **Step 8: Round to nearest tenth** 14.1083 → 14.1 --- **Final Answer:** The radius must be **14.1 cm** (to the nearest tenth).

2. A rectangular prism has dimensions 15 cm × 10 cm × 8 cm. Calculate its volume and total surface area. (Format: Volume = ? cm³, Surface Area = ? cm²) Answer: Volume = 1200 cm³, Surface Area = 700 cm² Solution: Calculate the volume using the formula V = l × w × h V = 15 × 10 × 8 V = 150 × 8 V = 1200 cm³ Calculate the surface area using the formula SA = 2(lw + lh + wh) SA = 2(15×10 + 15×8 + 10×8) SA = 2(150 + 120 + 80) SA = 2(350) SA = 700 cm² The answer is Volume = 1200 cm³, Surface Area = 700 cm².
   Full step-by-step solution

   Step 1: Calculate the volume using the formula V = l × w × h V = 15 × 10 × 8 V = 150 × 8 V = 1200 cm³ Step 2: Calculate the surface area using the formula SA = 2(lw + lh + wh) SA = 2(15×10 + 15×8 + 10×8) SA = 2(150 + 120 + 80) SA = 2(350) SA = 700 cm² The answer is Volume = 1200 cm³, Surface Area = 700 cm².

3. Mere is painting a cylindrical storage tank for her school's art project. The tank has a radius of 14 cm and a height of 30 cm. She needs to know the volume of the tank to fill it with paint, and the surface area to determine how much wrapping paper she needs to cover the entire tank (including the top and bottom). Find both the volume and the total surface area of the cylinder. Use π ≈ 22/7. Answer: Volume = 18480 cubic cm, Surface Area = 3872 square cm Solution: Write the formulas. Volume of cylinder = π × r² × h Surface area of cylinder = 2πr² + 2πrh (two circles plus the lateral rectangle) Substitute r = 14 cm, h = 30 cm, π = 22/7.
   Full step-by-step solution

   Step 1: Write the formulas. Volume of cylinder = π × r² × h Surface area of cylinder = 2πr² + 2πrh (two circles plus the lateral rectangle) Step 2: Substitute r = 14 cm, h = 30 cm, π = 22/7. Volume: π × r² × h = (22/7) × (14 × 14) × 30 First, 14 × 14 = 196 Then (22/7) × 196 = 22 × 28 = 616 (since 196 ÷ 7 = 28) Now 616 × 30 = 18480 Volume = 18480 cubic cm Surface Area: 2πr² = 2 × (22/7) × 196 = 2 × 22 × 28 = 2 × 616 = 1232 2πrh = 2 × (22/7) × 14 × 30 = 2 × 22 × 2 × 30 (since 14 ÷ 7 = 2) = 2 × 22 × 60 = 2 × 1320 = 2640 Total surface area = 1232 + 2640 = 3872 square cm Final answer: Volume = 18480 cubic cm, Surface Area = 3872 square cm

4. Ava is creating a triangular prism-shaped display case for her art project. The triangular base has a base length of 12 inches and a height of 9 inches. The length (depth) of the prism is 15 inches. What is the total volume and surface area of the display case? (Round the surface area to the nearest whole square inch if necessary.) Answer: Volume: 810 cubic inches; Surface Area: 648 square inches Solution: Find the area of the triangular base. Area of triangle = (1/2) * base * height = (1/2) * 12 * 9 = 54 square inches. Volume of the prism = base area * depth = 54 * 15 = 810 cubic inches.
   Full step-by-step solution

   Step 1: Find the area of the triangular base. Area of triangle = (1/2) * base * height = (1/2) * 12 * 9 = 54 square inches. Step 2: Volume of the prism = base area * depth = 54 * 15 = 810 cubic inches. Step 3: To find surface area, we need the perimeter of the triangle. The triangle has base = 12 inches and height = 9 inches. Since the height is drawn to the base, it splits the base into two equal segments of 6 inches each (assuming the triangle is isosceles). The two equal sides are found using the Pythagorean theorem: side = sqrt(6^2 + 9^2) = sqrt(36 + 81) = sqrt(117) ≈ 10.8167 inches. So the triangle has sides: 12 inches, 10.8167 inches, and 10.8167 inches. Step 4: Perimeter of triangle = 12 + 10.8167 + 10.8167 = 33.6334 inches. Step 5: Surface area = 2 * (base area) + (perimeter of base) * (depth) = 2 * 54 + 33.6334 * 15 = 108 + 504.501 = 612.501 square inches. Step 6: Wait — the prism has 5 faces: two triangular ends (2 * 54 = 108) and three rectangular faces. The rectangular faces have dimensions: 12 x 15 = 180, and two rectangles of 10.8167 x 15 = 162.2505 each. So total area of rectangles = 180 + 162.2505 + 162.2505 = 504.501. Add the two triangles: 108 + 504.501 = 612.501. Rounded to nearest whole number: 613 square inches. Step 7: However, re-reading the problem: the triangle's base and height are given, but we assumed isosceles. For a general triangle with base 12 and height 9, the side lengths are not uniquely determined unless the triangle is right or isosceles. Since the height is given as the perpendicular height, the most common assumption in Grade 7 problems is that the triangle is a right triangle where the base and height are the legs. In that case: the triangle has legs 12 and 9, so the hypotenuse = sqrt(12^2 + 9^2) = sqrt(144 + 81) = sqrt(225) = 15 inches. Step 8: Then perimeter = 12 + 9 + 15 = 36 inches. Step 9: Surface area = 2 * 54 + 36 * 15 = 108 + 540 = 648 square inches. Step 10: Final answer: Volume = 810 cubic inches, Surface Area = 648 square inches.

5. ∛(125) + √(144) = ? Answer: 17 Solution: We need to evaluate the cube root of 125 and the square root of 144, then add them together. Evaluate the cube root of 125.
   Full step-by-step solution

   Step 1: Understand the problem. We need to evaluate the cube root of 125 and the square root of 144, then add them together. Step 2: Evaluate the cube root of 125. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. We ask: what number multiplied by itself three times equals 125? 5 × 5 × 5 = 25 × 5 = 125. So, ∛(125) = 5. Step 3: Evaluate the square root of 144. The square root of a number is a value that, when multiplied by itself, gives the original number. We ask: what number multiplied by itself equals 144? 12 × 12 = 144. So, √(144) = 12. Step 4: Add the two results. 5 + 12 = 17. Step 5: Final answer. The result of ∛(125) + √(144) is 17.