Evaluate Roots

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Answer Key & Explanations

Evaluate Roots · Grade 8 · Worksheet 3

1. A cylindrical water tank has a diameter of 10 meters and a height of 8 meters. The tank is currently filled to 60% of its capacity. What is the volume of water in the tank in cubic meters? (Use π = 3.14) Answer: 376.8 Solution: Radius = diameter ÷ 2 = 10 ÷ 2 = 5 meters Volume = π × radius² × height Volume = 3.14 × 5² × 8 Volume = 3.14 × 25 × 8 Volume = 3.14 × 200 Volume = 628 cubic meters Calculate the volume of water at 60% capacity Water volume = 60% of total volume Water volume = 0.60 × 628 Water volume = 376.8…
   Full step-by-step solution

   Step 1: Find the radius of the cylinder Radius = diameter ÷ 2 = 10 ÷ 2 = 5 meters Step 2: Calculate the total volume of the cylinder Volume = π × radius² × height Volume = 3.14 × 5² × 8 Volume = 3.14 × 25 × 8 Volume = 3.14 × 200 Volume = 628 cubic meters Step 3: Calculate the volume of water at 60% capacity Water volume = 60% of total volume Water volume = 0.60 × 628 Water volume = 376.8 cubic meters The answer is 376.8.

2. Mason is helping his school design a new square playground that will have an area of 1,225 square meters. The school also wants to build a cubic storage shed for sports equipment with a volume of exactly 1,331 cubic meters. What is the side length of the square playground in meters, and what is the edge length of the cubic storage shed in meters? Answer: 35 and 11 Solution: Find the side length of the square playground. The area of a square is side × side = side². Area = 1,225 square meters.
   Full step-by-step solution

   Step 1: Find the side length of the square playground. The area of a square is side × side = side². Area = 1,225 square meters. We need to find √1,225. Since 30² = 900 and 40² = 1,600, the square root is between 30 and 40. Try 35 × 35 = 1,225. So √1,225 = 35. The side length of the playground is 35 meters. Step 2: Find the edge length of the cubic storage shed. The volume of a cube is edge × edge × edge = edge³. Volume = 1,331 cubic meters. We need to find ∛1,331. Since 10³ = 1,000 and 11³ = 1,331, the cube root is 11. Check: 11 × 11 = 121, and 121 × 11 = 1,331. So ∛1,331 = 11. The edge length of the shed is 11 meters. Step 3: State both answers. The square playground has a side length of 35 meters, and the cubic storage shed has an edge length of 11 meters.

3. Emma is designing a community garden with a square plot that has an area of 324 square meters. She wants to build a cubic rainwater collection tank where the volume is exactly 1,728 cubic meters. What is the side length of the square garden, and what is the edge length of the cubic tank? Answer: 18 and 12 Solution: Find the side length of the square garden. The area of a square is side × side = side². So side = sqrt(area) = sqrt(324).
   Full step-by-step solution

   Step 1: Find the side length of the square garden. The area of a square is side × side = side². So side = sqrt(area) = sqrt(324). Since 18 × 18 = 324, the side length of the garden is 18 meters. Step 2: Find the edge length of the cubic tank. The volume of a cube is edge × edge × edge = edge³. So edge = cube root of volume = cube root(1,728). Since 12 × 12 × 12 = 1,728, the edge length of the tank is 12 meters. Step 3: The side length of the garden is 18 meters and the edge length of the tank is 12 meters.

4. Liam is designing a cubic container to hold exactly 729 cubic centimeters of liquid. He needs to know the exact side length of the cube. What is the side length, in centimeters, of Liam's container? Answer: 9 Solution: We are told the container is a cube with volume 729 cubic centimeters. Let the side length be \( s \) cm. Write the formula for the volume of a cube.
   Full step-by-step solution

   We are told the container is a cube with volume 729 cubic centimeters. Let the side length be \( s \) cm. Step 1: Write the formula for the volume of a cube. Volume = side × side × side = \( s^3 \). Step 2: Set up the equation. \( s^3 = 729 \). Step 3: We need to find \( s \), which is the cube root of 729. We can try to factor 729 into smaller numbers to find its cube root. Step 4: Factor 729 step by step. 729 ÷ 3 = 243 243 ÷ 3 = 81 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 So 729 = 3 × 3 × 3 × 3 × 3 × 3. Step 5: Group the factors in triples for cube root. (3 × 3) × (3 × 3) × (3 × 3) is not the right grouping — instead: 3 × 3 × 3 = 27, but let's group directly: 729 = (3 × 3) × (3 × 3) × (3 × 3) = 9 × 9 × 9. So \( 729 = 9^3 \). Step 6: Therefore, \( s = 9 \). Step 7: Check: \( 9 \times 9 \times 9 = 81 \times 9 = 729 \), correct. Final answer: The side length is 9 centimeters.

5. Liam is designing a community garden with a square plot that has an area of 289 square meters. He wants to build a cubic storage container for gardening supplies that has the same side length as the garden plot. What is the volume, in cubic meters, of the storage container Liam is planning to build? Answer: 4913 Solution: Find the side length of the square garden plot. The area of the square is 289 square meters. For a square, area = side length × side length = side length squared.
   Full step-by-step solution

   Step 1: Find the side length of the square garden plot. The area of the square is 289 square meters. For a square, area = side length × side length = side length squared. So, side length squared = 289. Taking the square root: side length = √289. Since 17 × 17 = 289, the side length is 17 meters. Step 2: The storage container is a cube with the same side length as the garden plot. So, the side length of the cube = 17 meters. Step 3: Find the volume of the cube. Volume of a cube = side length × side length × side length. So, volume = 17 × 17 × 17. Step 4: Calculate step by step. First, 17 × 17 = 289. Then, 289 × 17 = 289 × (10 + 7) = 289 × 10 + 289 × 7. 289 × 10 = 2890. 289 × 7 = 2023. Add: 2890 + 2023 = 4913. Step 5: State the final answer. The volume of the storage container is 4913 cubic meters.

6. A scientist is studying the growth of a special type of algae. The volume of algae doubles every hour. After t hours, the volume is given by V(t) = 125 × 2^t cubic millimeters. After how many hours will the volume reach exactly 4000 cubic millimeters? Answer: 5 Solution: Set up the equation: 125 × 2^t = 4000 Divide both sides by 125: 2^t = 4000 ÷ 125 Calculate 4000 ÷ 125 = 32 Now we have 2^t = 32 Recognize that 32 is a power of 2: 32 = 2^5 Therefore, 2^t = 2^5 Since the bases are equal, the exponents must be equal: t = 5 The answer is 5.
   Full step-by-step solution

   Step 1: Set up the equation: 125 × 2^t = 4000 Step 2: Divide both sides by 125: 2^t = 4000 ÷ 125 Step 3: Calculate 4000 ÷ 125 = 32 Step 4: Now we have 2^t = 32 Step 5: Recognize that 32 is a power of 2: 32 = 2^5 Step 6: Therefore, 2^t = 2^5 Step 7: Since the bases are equal, the exponents must be equal: t = 5 The answer is 5.