3D Cross Sections

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Answer Key & Explanations

3D Cross Sections · Grade 6 · Worksheet 3

1. Emma is a chef preparing a special dessert. She has a block of cheese shaped like a rectangular prism that measures 15 inches long, 11 inches wide, and 7 inches tall. For her recipe, she needs to make a horizontal cut through the block at a height of 3 inches from the bottom. What is the area of the rectangular cross-section she creates with this cut? Answer: 165 Solution: A horizontal cut through a rectangular prism, parallel to the base, creates a cross-section that is a rectangle. The length of the cross-section is 15 inches, and the width is 11 inches.
   Full step-by-step solution

   Step 1: A horizontal cut through a rectangular prism, parallel to the base, creates a cross-section that is a rectangle. Step 2: The cross-section has the same length and width as the base of the prism, regardless of the height of the cut. Step 3: The length of the cross-section is 15 inches, and the width is 11 inches. Step 4: Calculate the area: Area = length × width = 15 × 11 Step 5: 15 × 11 = 165 Step 6: The area of the cross-section is 165 square inches.

2. A rectangular prism has dimensions of 12 cm by 8 cm by 5 cm. If you make a horizontal cut through the prism parallel to its base, creating two smaller prisms, and the height of the top prism is 2 cm, what is the volume of the bottom prism in cubic centimeters? Answer: 288 Solution: - Length = 12 cm - Width = 8 cm - Height = 5 cm A horizontal cut is made parallel to the base, creating two smaller prisms. The top prism's height is 2 cm. We need the volume of the bottom prism.
   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, creating two smaller prisms. The top prism's height is 2 cm. We need the volume of the bottom prism. --- **Step 2: Determine the height of the bottom prism** Total height = 5 cm Top prism height = 2 cm Bottom prism height = Total height − Top prism height = 5 cm − 2 cm = 3 cm --- **Step 3: Volume formula for a rectangular prism** Volume = Length × Width × Height --- **Step 4: Apply to bottom prism** Bottom prism has same length and width as original prism (since cut is parallel to base). Length = 12 cm Width = 8 cm Height = 3 cm Volume of bottom prism = 12 × 8 × 3 --- **Step 5: Calculate** 12 × 8 = 96 96 × 3 = 288 --- **Step 6: Final answer** Volume of bottom prism = 288 cubic centimeters. --- **Answer:** 288

3. (-15) + 28 - (-7) = ? Answer: 20 Solution: (-15) + 28 - (-7) Subtracting a negative is the same as adding a positive: - (-7) = + 7 (-15) + 28 + 7 First: (-15) + 28 Think of it as 28 - 15 = 13 So now we have: 13 + 7 13 + 7 = 20 Final Answer: 20
   Full step-by-step solution

   Let's solve step-by-step: Step 1: Write the problem clearly (-15) + 28 - (-7) Step 2: Interpret the subtraction of a negative number Subtracting a negative is the same as adding a positive: - (-7) = + 7 So the expression becomes: (-15) + 28 + 7 Step 3: Add the numbers from left to right First: (-15) + 28 Think of it as 28 - 15 = 13 So now we have: 13 + 7 Step 4: Add the last number 13 + 7 = 20 Final Answer: 20

4. (-18) × 2 + 36 ÷ (-4) = ? Answer: -45 Solution: First, perform the multiplication: (-18) × 2 = -36 Next, perform the division: 36 ÷ (-4) = -9 Now add the results: -36 + (-9) = -45 The answer is -45.
   Full step-by-step solution

   Step 1: First, perform the multiplication: (-18) × 2 = -36 Step 2: Next, perform the division: 36 ÷ (-4) = -9 Step 3: Now add the results: -36 + (-9) = -45 The answer is -45.

5. Emma slices a rectangular prism that has a length of 20 cm, a width of 15 cm, and a height of 10 cm. She makes a horizontal cut halfway up the height. What is the area of the resulting cross section in square centimeters? Answer: 300 Solution: A horizontal cut through a rectangular prism creates a rectangle that is parallel to the base. Step 2: The cross section will have the same length and width as the base of the prism.
   Full step-by-step solution

   Step 1: A horizontal cut through a rectangular prism creates a rectangle that is parallel to the base. Step 2: The cross section will have the same length and width as the base of the prism. Step 3: The length is 20 cm and the width is 15 cm. Step 4: The area of a rectangle is length times width. Step 5: Area = 20 cm × 15 cm = 300 square cm. The answer is 300.