Evaluate Roots

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Answer Key & Explanations

Evaluate Roots · Grade 8 · Worksheet 2

1. Isabella is helping her school design a new playground. The school wants a square sandbox with an area of 441 square feet. They also want to build a cube-shaped storage box for playground balls that has a volume of 729 cubic feet. What is the side length of the square sandbox and the edge length of the cube-shaped storage box? Answer: 21 feet and 9 feet Solution: Find the side length of the square sandbox. The area of a square is side squared (side × side). We know the area is 441 square feet, so we need to find the square root of 441.
   Full step-by-step solution

   Step 1: Find the side length of the square sandbox. The area of a square is side squared (side × side). We know the area is 441 square feet, so we need to find the square root of 441. Think: 21 × 21 = 441. So √441 = 21. The side length is 21 feet. Step 2: Find the edge length of the cube-shaped storage box. The volume of a cube is edge cubed (edge × edge × edge). We know the volume is 729 cubic feet, so we need to find the cube root of 729. Think: 9 × 9 × 9 = 81 × 9 = 729. So ∛729 = 9. The edge length is 9 feet. Step 3: State both answers. The square sandbox side length is 21 feet, and the cube-shaped storage box edge length is 9 feet.

2. ∛(512) + √(256) = ? Answer: 24 Solution: Find the cube root of 512. Since 8 × 8 × 8 = 512, ∛(512) = 8 Find the square root of 256. Since 16 × 16 = 256, √(256) = 16 Add the results: 8 + 16 = 24 The answer is 24.
   Full step-by-step solution

   Step 1: Find the cube root of 512. Since 8 × 8 × 8 = 512, ∛(512) = 8 Step 2: Find the square root of 256. Since 16 × 16 = 256, √(256) = 16 Step 3: Add the results: 8 + 16 = 24 The answer is 24.

3. A cylindrical water tank has a diameter of 10 meters and a height of 8 meters. The tank is filled to 70% of its capacity. What is the volume of water in the tank in cubic meters? (Use π = 3.14) Answer: 439.6 Solution: Radius = diameter ÷ 2 = 10 ÷ 2 = 5 meters Volume = π × radius² × height = 3.14 × (5)² × 8 = 3.14 × 25 × 8 = 3.14 × 200 = 628 cubic meters Water volume = 70% of full volume = 0.70 × 628 = 0.70 × 628 = 439.6 cubic meters The answer is 439.6.
   Full step-by-step solution

   Step 1: Find the radius of the cylinder Radius = diameter ÷ 2 = 10 ÷ 2 = 5 meters Step 2: Calculate the full volume of the cylinder Volume = π × radius² × height = 3.14 × (5)² × 8 = 3.14 × 25 × 8 = 3.14 × 200 = 628 cubic meters Step 3: Calculate the volume of water Water volume = 70% of full volume = 0.70 × 628 = 0.70 × 628 = 439.6 cubic meters The answer is 439.6.

4. A rectangular prism has a length of 12 cm, a width of 8 cm, and a height of 5 cm. If you need to paint all exterior surfaces of this prism, what is the total surface area in square centimeters? Answer: 392 Solution: A rectangular prism has 6 faces, but opposite faces are equal in area. Length (L) = 12 cm Width (W) = 8 cm Height (H) = 5 cm Identify the areas of each pair of opposite faces.
   Full step-by-step solution

   Let's find the total surface area of the rectangular prism. Step 1: Understand the problem. A rectangular prism has 6 faces, but opposite faces are equal in area. We have: Length (L) = 12 cm Width (W) = 8 cm Height (H) = 5 cm Step 2: Identify the areas of each pair of opposite faces. - Front and back faces: Each is a rectangle with dimensions L × H. Area of one front/back face = 12 × 5 = 60 cm² There are 2 such faces, so total = 2 × 60 = 120 cm² - Left and right faces: Each is a rectangle with dimensions W × H. Area of one left/right face = 8 × 5 = 40 cm² There are 2 such faces, so total = 2 × 40 = 80 cm² - Top and bottom faces: Each is a rectangle with dimensions L × W. Area of one top/bottom face = 12 × 8 = 96 cm² There are 2 such faces, so total = 2 × 96 = 192 cm² Step 3: Add all these totals together. Total surface area = 120 + 80 + 192 Step 4: Perform the addition. 120 + 80 = 200 200 + 192 = 392 Step 5: State the final answer. The total surface area to be painted is 392 cm². Answer: 392