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Evaluate Roots

Grade 8 · Algebra · Worksheet 1

  1. Noah is designing a square mural for the school art show. The area of the mural must be 484 square feet. He also wants to build a cube-shaped display pedestal for a sculpture, with a volume of exactly 729 cubic feet. What are the side length of the square mural and the edge length of the cube-shaped pedestal?
    Answer: ______________
  2. Mason is building a square patio in his backyard. The area of the patio needs to be 289 square feet. He also wants to build a cube-shaped planter box next to the patio. The volume of the planter box must be 343 cubic feet. What are the side length of the square patio and the edge length of the cube-shaped planter box?
    Answer: ______________
  3. Tane is designing a square mural for the school gymnasium wall. The mural must have an area of 625 square feet. He also wants to build a cube-shaped storage locker for art supplies that has a volume of 1,331 cubic feet. What is the side length of the square mural, in feet, and the edge length of the cube-shaped storage locker, in feet?
    Answer: ______________
  4. Liam is designing a cubic storage container that needs to hold exactly 64 cubic meters of water. He wants to calculate the exact side length of this cube. What is the side length in meters? Answer: ______________
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Answer Key & Explanations

Evaluate Roots · Grade 8 · Worksheet 1

  1. Noah is designing a square mural for the school art show. The area of the mural must be 484 square feet. He also wants to build a cube-shaped display pedestal for a sculpture, with a volume of exactly 729 cubic feet. What are the side length of the square mural and the edge length of the cube-shaped pedestal? Answer: 22 and 9 Solution: Find the side length of the square mural. The area of a square is side squared. So side = sqrt(484).
    Full step-by-step solution

    Step 1: Find the side length of the square mural. The area of a square is side squared. So side = sqrt(484). Since 22 * 22 = 484, sqrt(484) = 22. The side length of the mural is 22 feet. Step 2: Find the edge length of the cube-shaped pedestal. The volume of a cube is edge cubed. So edge = cube root of 729. Since 9 * 9 * 9 = 729, the cube root of 729 is 9. The edge length of the pedestal is 9 feet. Step 3: State both answers. The mural side length is 22 feet and the pedestal edge length is 9 feet.

  2. Mason is building a square patio in his backyard. The area of the patio needs to be 289 square feet. He also wants to build a cube-shaped planter box next to the patio. The volume of the planter box must be 343 cubic feet. What are the side length of the square patio and the edge length of the cube-shaped planter box? Answer: 17 feet and 7 feet Solution: Find the side length of the square patio. The area of a square is side squared. So side = sqrt(area).
    Full step-by-step solution

    Step 1: Find the side length of the square patio. The area of a square is side squared. So side = sqrt(area). sqrt(289) = 17 because 17 x 17 = 289. The side length of the patio is 17 feet. Step 2: Find the edge length of the cube-shaped planter box. The volume of a cube is edge cubed. So edge = cube root of volume. cube root of 343 = 7 because 7 x 7 x 7 = 343. The edge length of the planter box is 7 feet. Final answer: The side length of the square patio is 17 feet and the edge length of the cube-shaped planter box is 7 feet.

  3. Tane is designing a square mural for the school gymnasium wall. The mural must have an area of 625 square feet. He also wants to build a cube-shaped storage locker for art supplies that has a volume of 1,331 cubic feet. What is the side length of the square mural, in feet, and the edge length of the cube-shaped storage locker, in feet? Answer: 25 and 11 Solution: Find the side length of the square mural. The area of a square is side squared. We need the square root of 625.
    Full step-by-step solution

    Step 1: Find the side length of the square mural. The area of a square is side squared. We need the square root of 625. Since 25 x 25 = 625, the side length is 25 feet. Step 2: Find the edge length of the cube-shaped storage locker. The volume of a cube is edge cubed. We need the cube root of 1331. Since 11 x 11 x 11 = 1331, the edge length is 11 feet. Step 3: State both answers. The square mural has a side length of 25 feet, and the cube-shaped storage locker has an edge length of 11 feet.

  4. Liam is designing a cubic storage container that needs to hold exactly 64 cubic meters of water. He wants to calculate the exact side length of this cube. What is the side length in meters? Answer: 4 Solution: We are told the container is a cube and holds 64 cubic meters of water. That means the volume of the cube is 64 m³. Recall the formula for the volume of a cube.
    Full step-by-step solution

    We are told the container is a cube and holds 64 cubic meters of water. That means the volume of the cube is 64 m³. Step 1: Recall the formula for the volume of a cube. Volume = side length × side length × side length If we call the side length \( s \), then: Volume = s³ Step 2: Substitute the given volume into the formula. s³ = 64 Step 3: Find the side length \( s \) by taking the cube root of both sides. s = cube root of 64 Step 4: Determine what number multiplied by itself three times gives 64. Check possible whole numbers: 2³ = 8 (too small) 3³ = 27 (too small) 4³ = 4 × 4 × 4 = 16 × 4 = 64 (correct) Step 5: Conclusion. The side length is 4 meters. Final answer: 4