Evaluate Roots

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Answer Key & Explanations

Evaluate Roots · Grade 8 · Worksheet 2

1. Kaia is an artist designing a square canvas for her next painting. The area of the canvas is 529 square inches. She also wants to create a small cubic sculpture to place beside the canvas. The volume of the cube is 343 cubic inches. What is the side length of the square canvas in inches, and what is the edge length of the cubic sculpture in inches? Answer: 23 and 7 Solution: Find the side length of the square canvas. The area of a square is side squared, so we need the square root of 529. Since 23 × 23 = 529, the side length is 23 inches.
   Full step-by-step solution

   Step 1: Find the side length of the square canvas. The area of a square is side squared, so we need the square root of 529. Since 23 × 23 = 529, the side length is 23 inches. Step 2: Find the edge length of the cubic sculpture. The volume of a cube is edge cubed, so we need the cube root of 343. Since 7 × 7 × 7 = 343, the edge length is 7 inches. Step 3: The side length of the square canvas is 23 inches, and the edge length of the cubic sculpture is 7 inches.

2. 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.

3. A cylindrical water tank has a diameter of 12 meters and a height of 8 meters. The tank is currently filled to 75% of its capacity. What is the volume of water in the tank in cubic meters? (Use π = 3.14) Answer: 678.24 Solution: Radius = Diameter ÷ 2 = 12 ÷ 2 = 6 meters Volume = π × radius² × height Volume = 3.14 × 6² × 8 Volume = 3.14 × 36 × 8 Volume = 3.14 × 288 Volume = 904.32 cubic meters Calculate the volume of water at 75% capacity Water volume = 75% of full volume Water volume = 0.75 × 904.32 Water volume =…
   Full step-by-step solution

   Step 1: Find the radius of the cylinder Radius = Diameter ÷ 2 = 12 ÷ 2 = 6 meters Step 2: Calculate the full volume of the cylinder Volume = π × radius² × height Volume = 3.14 × 6² × 8 Volume = 3.14 × 36 × 8 Volume = 3.14 × 288 Volume = 904.32 cubic meters Step 3: Calculate the volume of water at 75% capacity Water volume = 75% of full volume Water volume = 0.75 × 904.32 Water volume = 678.24 cubic meters The answer is 678.24.

4. ∛(512) + √(225) = ? Answer: 23 Solution: Find the cube root of 512. Since 8 × 8 × 8 = 512, ∛(512) = 8 Find the square root of 225. Since 15 × 15 = 225, √(225) = 15 Add the results: 8 + 15 = 23 The answer is 23.
   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 225. Since 15 × 15 = 225, √(225) = 15 Step 3: Add the results: 8 + 15 = 23 The answer is 23.