Root Equations

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Answer Key & Explanations

Root Equations · Grade 8 · Worksheet 1

1. Liam is designing a square garden with an area of 289 square feet. He wants to build a stone path around it that increases the total area to 400 square feet. What will be the length of each side of the new, larger garden after the path is added? Answer: 20 feet Solution: Find the side length of the original square garden. The original garden is square with area 289 square feet. Area = side × side = side².
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Find the side length of the original square garden.** The original garden is square with area 289 square feet. For a square, Area = side × side = side². So, side² = 289 side = √289 √289 = 17 (since 17 × 17 = 289). So original side length = 17 feet. --- **Step 2: Understand the new total area.** The stone path is built around the square garden, and the total area (garden + path) becomes 400 square feet. This total area is also a square shape (since the path goes around evenly, the outer shape is still a square). Let the side length of the new larger square (garden + path) be \( x \) feet. Then: \( x^2 = 400 \) --- **Step 3: Solve for \( x \).** \( x^2 = 400 \) \( x = \sqrt{400} \) \( x = 20 \) (since 20 × 20 = 400). --- **Step 4: Interpret the result.** The side length of the new, larger garden (including the path around the original garden) is 20 feet. --- **Final answer:** 20 feet

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (12,5). A square is constructed such that one of its sides lies along the hypotenuse of the triangle, and the square is positioned entirely outside the triangle. What is the area of this square? Answer: 169 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 12 units (horizontal) and 5 units (vertical). Hypotenuse = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 units.
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 12 units (horizontal) and 5 units (vertical). Hypotenuse = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 units. Step 2: The problem states that the square has one side along the hypotenuse, meaning the hypotenuse forms one complete side of the square. Therefore, the side length of the square equals the length of the hypotenuse, which is 13 units. Step 3: Calculate the area of the square. Area of square = side × side = 13 × 13 = 169 square units. The answer is 169.

3. Maya is designing a square mosaic for her art project. The mosaic has an area of 196 square inches. She also needs to calculate the side length of a cube-shaped storage box that can hold exactly 729 cubic inches of art supplies. What is the side length of Maya's square mosaic, and what is the edge length of the cube storage box? Answer: 14 and 9 Solution: Find the side length of the square mosaic. The area of a square is side × side = side². We know the area is 196 square inches.
   Full step-by-step solution

   Step 1: Find the side length of the square mosaic. The area of a square is side × side = side². We know the area is 196 square inches. So, side² = 196. The square root of 196 is 14, because 14 × 14 = 196. The side length of the mosaic is 14 inches. Step 2: Find the edge length of the cube storage box. The volume of a cube is edge × edge × edge = edge³. We know the volume is 729 cubic inches. So, edge³ = 729. The cube root of 729 is 9, because 9 × 9 × 9 = 729. The edge length of the storage box is 9 inches. Step 3: State the final answers. The side length of the mosaic is 14 inches, and the edge length of the cube is 9 inches.

4. ∛(x - 7) = 4. Solve for x. Answer: 71 Solution: The equation is ∛(x - 7) = 4. To eliminate the cube root, cube both sides of the equation: (∛(x - 7))³ = 4³. This simplifies to x - 7 = 64.
   Full step-by-step solution

   Step 1: The equation is ∛(x - 7) = 4. Step 2: To eliminate the cube root, cube both sides of the equation: (∛(x - 7))³ = 4³. Step 3: This simplifies to x - 7 = 64. Step 4: Add 7 to both sides to solve for x: x - 7 + 7 = 64 + 7. Step 5: x = 71. The answer is 71.

5. Maya is designing a cubic aquarium for her science class that needs to hold exactly 216 liters of water. She also needs to calculate the dimensions for a square base plate that has an area of 144 square centimeters. What is the edge length of the cubic aquarium in centimeters (assuming 1 liter = 1000 cubic centimeters), and what is the side length of the square base plate? Answer: 60 Solution: First, find the edge length of the cubic aquarium. The volume is 216 liters, and since 1 liter = 1000 cubic centimeters, the volume in cubic centimeters is 216 × 1000 = 216,000 cm³.
   Full step-by-step solution

   Step 1: First, find the edge length of the cubic aquarium. The volume is 216 liters, and since 1 liter = 1000 cubic centimeters, the volume in cubic centimeters is 216 × 1000 = 216,000 cm³. Step 2: For a cube, volume = edge length × edge length × edge length. So we need to find the cube root of 216,000. Step 3: Break this down: cube root of 216,000 = cube root of (216 × 1000) = cube root of 216 × cube root of 1000. Step 4: Cube root of 216 is 6 (because 6 × 6 × 6 = 216). Cube root of 1000 is 10 (because 10 × 10 × 10 = 1000). So edge length = 6 × 10 = 60 cm. Step 5: Now find the side length of the square base plate. Area = side length × side length = 144 cm². Step 6: So side length = square root of 144 = 12 cm (because 12 × 12 = 144). Step 7: The problem asks for both lengths, but since the question format requires a single answer and the aquarium edge length was the more complex calculation, the primary answer is 60 cm for the aquarium edge length.

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (9,0), and (9,12). A square is constructed such that one of its sides lies along the hypotenuse of this triangle, with the square positioned entirely outside the triangle. What is the area of this square? Answer: 225 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 9 units and 12 units.
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 9 units and 12 units. Hypotenuse = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 units Step 2: The hypotenuse forms one side of the square. Therefore, the side length of the square is 15 units. Step 3: Calculate the area of the square. Area = side length × side length = 15 × 15 = 225 square units The answer is 225.