3D Cross Sections

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Answer Key & Explanations

3D Cross Sections · Grade 6 · Worksheet 2

1. A rectangular prism has dimensions of 12 cm by 8 cm by 5 cm. If you make a horizontal cut parallel to the base at a height of 3 cm from the bottom, what is the area of the cross-section you create? Answer: 96 Solution: Length = 12 cm Width = 8 cm Height = 5 cm A horizontal cut is made parallel to the base at a height of 3 cm from the bottom.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular prism with dimensions: Length = 12 cm Width = 8 cm Height = 5 cm A horizontal cut is made parallel to the base at a height of 3 cm from the bottom. --- **Step 2: Identify the cross-section shape** A horizontal cut parallel to the base means the cross-section is a rectangle with the same length and width as the base, because the base is a rectangle of 12 cm by 8 cm. --- **Step 3: Check if height affects the cross-section area** The height of the prism is 5 cm, but the cut is at 3 cm from the bottom. Since the cut is parallel to the base, the cross-section at that height will have the same length and width as the base, regardless of the height at which we cut (as long as it's between 0 and 5 cm). So: Length of cross-section = 12 cm Width of cross-section = 8 cm --- **Step 4: Calculate the area of the cross-section** Area = length × width Area = 12 × 8 Area = 96 cm² --- **Step 5: Final answer** The area of the cross-section is 96.

2. (-12 + 8) × (-3) = ? Answer: 12 Solution: First, handle the parentheses: (-12 + 8) -12 + 8 = -4 So now we have: (-4) × (-3) Multiply the two numbers: (-4) × (-3) When multiplying two negative numbers, the result is positive.
   Full step-by-step solution

   Let's solve step-by-step. Step 1: First, handle the parentheses: (-12 + 8) -12 + 8 = -4 So now we have: (-4) × (-3) Step 2: Multiply the two numbers: (-4) × (-3) When multiplying two negative numbers, the result is positive. 4 × 3 = 12 So (-4) × (-3) = 12 Step 3: Final answer: 12

3. Liam is building a rectangular prism-shaped planter box for his garden that measures 2.4 meters long, 1.2 meters wide, and 0.8 meters high. He wants to fill it with soil, but the soil is sold in bags that each contain 0.25 cubic meters. How many full bags of soil does Liam need to buy to completely fill the planter box? Answer: 10 Solution: First, find the volume of the planter box. Volume = length × width × height Length = 2.4 meters Width = 1.2 meters Height = 0.8 meters So, Volume = 2.4 × 1.2 × 0.8 Multiply length and width.
   Full step-by-step solution

   First, find the volume of the planter box. The box is a rectangular prism, so: Volume = length × width × height Given: Length = 2.4 meters Width = 1.2 meters Height = 0.8 meters So, Volume = 2.4 × 1.2 × 0.8 Step 1: Multiply length and width. 2.4 × 1.2 = 2.88 Step 2: Multiply the result by the height. 2.88 × 0.8 = 2.304 So, the volume of the planter box is 2.304 cubic meters. Now, each bag of soil contains 0.25 cubic meters. To find the number of bags needed, divide the total volume by the volume per bag: Number of bags = Total volume / Volume per bag Number of bags = 2.304 / 0.25 Step 3: Perform the division. Dividing by 0.25 is the same as multiplying by 4. So, 2.304 / 0.25 = 2.304 × 4 = 9.216 This means 9.216 bags are needed. Step 4: Interpret the result. The problem asks for the number of full bags needed to completely fill the box. Since 9 full bags would only provide 9 × 0.25 = 2.25 cubic meters, which is less than 2.304 cubic meters, 9 bags are not enough. Therefore, we must round up to the next whole number. 9.216 rounds up to 10 full bags. So, Liam needs to buy 10 full bags of soil. Final answer: 10

4. A cylindrical water tank has a diameter of 120 cm and a height of 250 cm. If you make a horizontal cut parallel to the circular base at a height of 100 cm from the bottom, what is the area of the circular cross-section created by this cut? (Use π = 3.14) Answer: 11304 cm² Solution: The cross-section is a circle parallel to the base, so its size depends on the cylinder's radius, not the height where it's cut. The diameter is 120 cm, so the radius is half of that: 120 ÷ 2 = 60 cm.
   Full step-by-step solution

   Step 1: The cross-section is a circle parallel to the base, so its size depends on the cylinder's radius, not the height where it's cut. Step 2: The diameter is 120 cm, so the radius is half of that: 120 ÷ 2 = 60 cm. Step 3: The area of a circle is calculated using the formula: Area = π × radius² Step 4: Substitute the values: Area = 3.14 × (60)² Step 5: Calculate (60)² = 60 × 60 = 3600 Step 6: Multiply: 3.14 × 3600 = 11304 Step 7: The area of the cross-section is 11304 cm².

5. Mason slices a cone with a height of 21 cm and a base radius of 14 cm with a horizontal cut 9 cm from the tip. What is the radius of the circular cross section? Answer: 6 cm Solution: The cone has height 21 cm and base radius 14 cm. A horizontal cut is made 9 cm from the tip. The distance from the tip to the cut is 9 cm.
   Full step-by-step solution

   Step 1: The cone has height 21 cm and base radius 14 cm. A horizontal cut is made 9 cm from the tip. Step 2: The distance from the tip to the cut is 9 cm. The total height is 21 cm. Step 3: Using similar triangles, the ratio of the cross section radius (r) to the base radius (14 cm) equals the ratio of the distance from the tip (9 cm) to the total height (21 cm). Step 4: Set up the proportion: r/14 = 9/21 Step 5: Simplify 9/21 = 3/7 Step 6: Multiply both sides by 14: r = 14 × (3/7) = 14 × 3 ÷ 7 = 42 ÷ 7 = 6 The radius of the circular cross section is 6 cm.