Pythagorean 3D

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Pythagorean 3D · Grade 8 · Worksheet 1

1. Maya is setting up a zip line between two trees in her backyard. The horizontal distance between the trees is 15 meters, and the height difference between the attachment points is 8 meters. If the zip line cable needs to be 2 meters longer than the direct diagonal distance to allow for sag and attachment, what total length of cable should Maya purchase? Round your answer to the nearest tenth of a meter. Answer: 19.0 Solution: Find the direct diagonal distance using the Pythagorean theorem Diagonal distance = sqrt(15^2 + 8^2) Diagonal distance = sqrt(225 + 64) Diagonal distance = sqrt(289) Diagonal distance = 17 meters Add the extra length needed for sag and attachment Total cable length = diagonal distance + extra…
   Full step-by-step solution

   Step 1: Find the direct diagonal distance using the Pythagorean theorem Diagonal distance = sqrt(15^2 + 8^2) Diagonal distance = sqrt(225 + 64) Diagonal distance = sqrt(289) Diagonal distance = 17 meters Step 2: Add the extra length needed for sag and attachment Total cable length = diagonal distance + extra length Total cable length = 17 + 2 Total cable length = 19 meters Step 3: Round to the nearest tenth 19.0 meters The answer is 19.0 meters.

2. Sophia is helping her school's drama club build a large rectangular storage trunk for costumes. The interior of the trunk is 30 cm long, 24 cm wide, and 18 cm high. She wants to place a straight metal rod from the bottom front left corner to the top back right corner of the trunk to use as a support for hanging costumes. What is the exact length, in centimeters, of the longest straight metal rod that can fit inside the trunk? Simplify the square root if possible. Answer: sqrt(1800) or 30*sqrt(2) Solution: The trunk is a rectangular prism with length = 30 cm, width = 24 cm, and height = 18 cm. The rod goes from the bottom front left corner to the top back right corner, which is the space diagonal.
   Full step-by-step solution

   Step 1: The trunk is a rectangular prism with length = 30 cm, width = 24 cm, and height = 18 cm. The rod goes from the bottom front left corner to the top back right corner, which is the space diagonal. Step 2: First, find the diagonal of the base (the floor). Use the Pythagorean theorem on the base rectangle: base diagonal^2 = length^2 + width^2 = 30^2 + 24^2 = 900 + 576 = 1476. So the base diagonal = sqrt(1476). Step 3: Simplify sqrt(1476). 1476 = 4 * 369 = 4 * 9 * 41 = 36 * 41. So sqrt(1476) = sqrt(36 * 41) = 6 * sqrt(41) cm. Step 4: Now consider the right triangle formed by the base diagonal (6*sqrt(41) cm), the height (18 cm), and the space diagonal (the rod). The space diagonal is the hypotenuse. Step 5: Apply the Pythagorean theorem again: space diagonal^2 = (base diagonal)^2 + height^2 = (6*sqrt(41))^2 + 18^2 = 36 * 41 + 324 = 1476 + 324 = 1800. Step 6: Take the square root: space diagonal = sqrt(1800). Simplify: 1800 = 100 * 18 = 100 * 9 * 2 = 900 * 2, so sqrt(1800) = sqrt(900 * 2) = 30 * sqrt(2). Step 7: The exact length of the longest straight metal rod is sqrt(1800) cm, or 30*sqrt(2) cm.

3. A rectangular prism 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 prism to the opposite corner? Round your answer to the nearest tenth of a centimeter. Answer: 14.1 Solution: We are finding the space diagonal of a rectangular prism with dimensions 6 cm, 8 cm, and 10 cm.
   Full step-by-step solution

   We are finding the space diagonal of a rectangular prism with dimensions 6 cm, 8 cm, and 10 cm. Step 1: Understand the formula for the space diagonal. For a rectangular prism with length l, width w, and height h, the space diagonal d is given by: d = sqrt(l^2 + w^2 + h^2) Step 2: Substitute the given dimensions. Here, l = 6 cm, w = 8 cm, h = 10 cm. So: d = sqrt(6^2 + 8^2 + 10^2) Step 3: Calculate each square. 6^2 = 36 8^2 = 64 10^2 = 100 Step 4: Add them. 36 + 64 + 100 = 200 Step 5: Take the square root. d = sqrt(200) Step 6: Simplify sqrt(200). 200 = 100 * 2, so sqrt(200) = sqrt(100 * 2) = sqrt(100) * sqrt(2) = 10 * sqrt(2) Step 7: Approximate sqrt(2) as 1.41421356... So 10 * sqrt(2) ≈ 10 * 1.41421356 ≈ 14.1421356 Step 8: Round to the nearest tenth. 14.1421356 rounded to the nearest tenth is 14.1 Final Answer: 14.1 cm

4. Mason is constructing a large rectangular wooden frame for a climbing wall in a community center. The frame has dimensions 18 feet by 24 feet by 30 feet. A support cable must run in a straight line from the top front left corner of the frame to the bottom back right corner. What is the length of this cable, to the nearest foot? Answer: 42 Solution: The frame is a rectangular prism with length = 24 ft, width = 18 ft, height = 30 ft. First, find the diagonal of the base rectangle (length and width).
   Full step-by-step solution

   Step 1: The frame is a rectangular prism with length = 24 ft, width = 18 ft, height = 30 ft. Step 2: First, find the diagonal of the base rectangle (length and width). Use the Pythagorean theorem: base diagonal = sqrt(24^2 + 18^2) = sqrt(576 + 324) = sqrt(900) = 30 ft. Step 3: The space diagonal (cable) forms a right triangle with the base diagonal (30 ft) and the height (30 ft). Step 4: Apply the Pythagorean theorem again: space diagonal = sqrt(30^2 + 30^2) = sqrt(900 + 900) = sqrt(1800). Step 5: Simplify sqrt(1800). 1800 = 100 * 18 = 100 * 9 * 2, so sqrt(1800) = sqrt(100) * sqrt(9) * sqrt(2) = 10 * 3 * sqrt(2) = 30 * sqrt(2). Step 6: sqrt(2) ≈ 1.4142, so 30 * 1.4142 = 42.426 ft. Round to the nearest foot: 42 ft. The cable is approximately 42 feet long.

5. √(12² + 16² + 21²) = ? Answer: 29 Solution: Square each dimension: 12² = 144, 16² = 256, 21² = 441 Add the squares: 144 + 256 + 441 = 841 Take the square root: √841 = 29 The answer is 29.
   Full step-by-step solution

   Step 1: Square each dimension: 12² = 144, 16² = 256, 21² = 441 Step 2: Add the squares: 144 + 256 + 441 = 841 Step 3: Take the square root: √841 = 29 The answer is 29.

6. √(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.