Pythagorean 3D

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Pythagorean 3D · Grade 8 · Worksheet 2

1. A rectangular box has dimensions of 6 cm by 8 cm by 10 cm. What is the length of the space diagonal that runs from one corner of the box to the opposite corner? Answer: 10√2 Solution: The space diagonal goes from one corner of the box to the farthest opposite corner. It passes through the inside of the box. If the sides have lengths a, b, and c, the space diagonal d is given by: d = sqrt(a^2 + b^2 + c^2) Identify the lengths.
   Full step-by-step solution

   Let's find the space diagonal of a box with dimensions 6 cm, 8 cm, and 10 cm. Step 1: Understand the problem. The space diagonal goes from one corner of the box to the farthest opposite corner. It passes through the inside of the box. Step 2: Recall the formula for the space diagonal of a rectangular prism. If the sides have lengths a, b, and c, the space diagonal d is given by: d = sqrt(a^2 + b^2 + c^2) Step 3: Identify the lengths. Here, a = 6 cm, b = 8 cm, c = 10 cm. Step 4: Apply the formula. First, calculate a^2 = 6^2 = 36 Then, b^2 = 8^2 = 64 Then, c^2 = 10^2 = 100 Step 5: Add the squares. a^2 + b^2 + c^2 = 36 + 64 + 100 = 200 Step 6: Take the square root. d = sqrt(200) Step 7: Simplify the square root. We can simplify sqrt(200). Notice that 200 = 100 * 2. So, sqrt(200) = sqrt(100 * 2) = sqrt(100) * sqrt(2) = 10 * sqrt(2) Step 8: Write the final answer. The length of the space diagonal is 10√2 cm. Therefore, the correct answer is 10√2.

2. Aroha is designing a large rectangular display case for a museum exhibit. The case has interior dimensions of 14 cm long, 11 cm wide, and 9 cm high. She wants to place a straight metal rod from the bottom front left corner to the top back right corner of the case to support a hanging artifact. What is the length of the longest rod that can fit in a straight line inside the case? Round your answer to the nearest tenth of a centimeter. Answer: 19.9 Solution: The rod runs from one corner to the opposite corner of the box. This is the space diagonal of a rectangular prism. Step 2: First, find the diagonal of the base (the floor).
   Full step-by-step solution

   Step 1: The rod runs from one corner to the opposite corner of the box. This is the space diagonal of a rectangular prism. Step 2: First, find the diagonal of the base (the floor). The base has length 14 cm and width 11 cm. Use the Pythagorean theorem: base diagonal = sqrt(14^2 + 11^2) = sqrt(196 + 121) = sqrt(317). Step 3: Now, this base diagonal and the height (9 cm) form a right triangle with the space diagonal as the hypotenuse. Step 4: Apply the Pythagorean theorem again: space diagonal = sqrt((sqrt(317))^2 + 9^2) = sqrt(317 + 81) = sqrt(398). Step 5: Calculate sqrt(398): 398 = 400 - 2, sqrt(398) is approximately 19.9499. Step 6: Round to the nearest tenth: 19.9 cm. The longest rod that fits is 19.9 cm.

3. A rectangular prism has dimensions of 6 cm by 8 cm by 10 cm. A spider needs to travel from one corner of the prism to the opposite corner along the shortest possible path on the surface. What is the length of this shortest path? Answer: √200 cm or 10√2 cm Solution: When finding the shortest path between two points on the surface of a 3D shape, we can often 'unfold' the shape to create a 2D surface.
   Full step-by-step solution

   When finding the shortest path between two points on the surface of a 3D shape, we can often 'unfold' the shape to create a 2D surface. The shortest path then becomes a straight line between the two points on this flat surface. This technique is particularly useful for rectangular prisms where we can consider different unfolding patterns to find the minimal distance.

4. √(6² + 8² + 10²) = ? Answer: 10√2 Solution: √(6² + 8² + 10²) 6² = 36 8² = 64 10² = 100 36 + 64 + 100 = 200 √(200) Factor 200 into prime factors 200 = 100 × 2 = (10²) × 2 √(200) = √(100 × 2) = √(100) × √(2) = 10 × √(2) 10√2
   Full step-by-step solution

   Let's solve step-by-step. Step 1: Write down the expression √(6² + 8² + 10²) Step 2: Calculate each square 6² = 36 8² = 64 10² = 100 Step 3: Add them 36 + 64 + 100 = 200 Step 4: Write the expression after addition √(200) Step 5: Factor 200 into prime factors 200 = 100 × 2 = (10²) × 2 Step 6: Take the square root √(200) = √(100 × 2) = √(100) × √(2) = 10 × √(2) Step 7: Final answer 10√2

5. √(8² + 15² + 20²) = ? Answer: 27 Solution: Square each dimension: 8² = 64, 15² = 225, 20² = 400 Sum the squares: 64 + 225 + 400 = 689 Take the square root: √689 = 27 (since 27 × 27 = 729, and 26 × 26 = 676, so 27 is the nearest whole number) The answer is 27.
   Full step-by-step solution

   Step 1: Square each dimension: 8² = 64, 15² = 225, 20² = 400 Step 2: Sum the squares: 64 + 225 + 400 = 689 Step 3: Take the square root: √689 = 27 (since 27 × 27 = 729, and 26 × 26 = 676, so 27 is the nearest whole number) The answer is 27.