Approximate Irrationals

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Answer Key & Explanations

Approximate Irrationals · Grade 8 · Worksheet 2

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). What is the length of the hypotenuse? Answer: 10 Solution: Identify the triangle's vertices. The vertices are at (0,0), (6,0), and (6,8). Determine which side is the hypotenuse.
   Full step-by-step solution

   Let's solve this step-by-step. Step 1: Identify the triangle's vertices. The vertices are at (0,0), (6,0), and (6,8). Step 2: Determine which side is the hypotenuse. Since it's a right triangle, the hypotenuse is the side opposite the right angle. The points (0,0) and (6,0) have the same y-coordinate, so that side is horizontal. The points (6,0) and (6,8) have the same x-coordinate, so that side is vertical. A horizontal line and a vertical line meet at a right angle, so the right angle is at (6,0). Therefore, the hypotenuse is the side between (0,0) and (6,8). Step 3: Use the distance formula to find the length of the hypotenuse. The distance formula between two points (x1, y1) and (x2, y2) is: distance = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ) Here, (x1, y1) = (0,0) and (x2, y2) = (6,8). Step 4: Substitute the coordinates into the formula. x2 - x1 = 6 - 0 = 6 y2 - y1 = 8 - 0 = 8 So, distance = sqrt( (6)^2 + (8)^2 ) Step 5: Calculate the squares. (6)^2 = 36 (8)^2 = 64 Step 6: Add the squares. 36 + 64 = 100 Step 7: Take the square root. sqrt(100) = 10 Step 8: Conclusion. The length of the hypotenuse is 10.

2. √(2) × √(18) = ? Answer: 6 Solution: Start with √(2) × √(18) Combine under one radical: √(2 × 18) Multiply inside the radical: √(36) Simplify the square root: √(36) = 6 The answer is 6.
   Full step-by-step solution

   Step 1: Start with √(2) × √(18) Step 2: Combine under one radical: √(2 × 18) Step 3: Multiply inside the radical: √(36) Step 4: Simplify the square root: √(36) = 6 The answer is 6.

3. Olivia is building a square-shaped mosaic for her art class. The area of the mosaic needs to be exactly 95 square inches. She wants to estimate the side length of the mosaic to the nearest tenth of an inch so she can cut the tiles accurately. What is the approximate side length of the mosaic, rounded to the nearest tenth of an inch? Answer: 9.7 inches Solution: The side length of a square is the square root of its area. So side length = sqrt(95) inches. Find the two consecutive whole numbers that sqrt(95) lies between.
   Full step-by-step solution

   Step 1: The side length of a square is the square root of its area. So side length = sqrt(95) inches. Step 2: Find the two consecutive whole numbers that sqrt(95) lies between. 9^2 = 81 and 10^2 = 100, so sqrt(95) is between 9 and 10. Step 3: Test tenths: 9.7^2 = 9.7 * 9.7 = 94.09, which is less than 95. 9.8^2 = 9.8 * 9.8 = 96.04, which is greater than 95. Step 4: Since 94.09 is closer to 95 than 96.04, sqrt(95) is approximately 9.7. Step 5: Therefore, the side length rounded to the nearest tenth is 9.7 inches. The answer is 9.7 inches.

4. Liam is designing a rectangular garden for his school's science project. The garden's length is √50 meters and its width is √18 meters. He needs to calculate the approximate area to determine how much soil to buy. What is the approximate area of the garden in square meters, rounded to the nearest whole number? Answer: 30 Solution: Area = length × width Length = √50 meters Width = √18 meters Area = √50 × √18 Area = √(50 × 18) 50 × 18 = 900 Area = √900 √900 = 30 So the exact area is 30 square meters. 30 is already a whole number. Final Answer: 30
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Write the formula for the area of a rectangle** Area = length × width Given: Length = √50 meters Width = √18 meters So: Area = √50 × √18 --- **Step 2: Combine the square roots** Area = √(50 × 18) --- **Step 3: Multiply inside the square root** 50 × 18 = 900 So: Area = √900 --- **Step 4: Simplify the square root** √900 = 30 So the exact area is 30 square meters. --- **Step 5: Round to the nearest whole number** 30 is already a whole number. --- **Final Answer:** 30

5. The area of a square garden is 150 square meters. Between which two consecutive whole numbers does the length of one side of the garden lie? Answer: 12 Solution: We are told the area of the square garden is 150 square meters. Recall the formula for the area of a square. Area = side × side, or A = s².
   Full step-by-step solution

   We are told the area of the square garden is 150 square meters. Step 1: Recall the formula for the area of a square. Area = side × side, or A = s². Step 2: Substitute the given area into the formula. s² = 150 Step 3: Find the side length by taking the square root of both sides. s = √150 Step 4: Simplify √150 if possible. √150 = √(25 × 6) = √25 × √6 = 5 × √6 Step 5: Estimate √6. We know 2² = 4 and 3² = 9, so √6 is between 2 and 3. More precisely: 2.4² = 5.76 2.45² = 6.0025 (very close to 6) So √6 ≈ 2.45 Step 6: Multiply by 5. s ≈ 5 × 2.45 = 12.25 Step 7: Determine between which two consecutive whole numbers 12.25 lies. 12 < 12.25 < 13 So the side length lies between 12 and 13. Final answer: 12

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (9,0), and (9,12). A circle is drawn such that its diameter is the hypotenuse of this triangle. What is the approximate area of the circle? Use π ≈ 3.14 and round your answer to the nearest whole number. Answer: 177 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs are 9 units and 12 units. Hypotenuse = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 units The hypotenuse is the diameter of the circle.
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs are 9 units and 12 units. Hypotenuse = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 units Step 2: The hypotenuse is the diameter of the circle. Diameter = 15 units Radius = diameter/2 = 15/2 = 7.5 units Step 3: Calculate the area of the circle. Area = π × radius^2 = 3.14 × (7.5)^2 = 3.14 × 56.25 = 176.625 Step 4: Round to the nearest whole number. 176.625 rounds to 177 The answer is 177.

7. √(55) ≈ ? (to the nearest tenth) Answer: 7.4 Solution: Identify perfect squares near 55. 7^2 = 49 and 8^2 = 64. So √55 is between 7 and 8.
   Full step-by-step solution

   Step 1: Identify perfect squares near 55. 7^2 = 49 and 8^2 = 64. So √55 is between 7 and 8. Step 2: Since 55 is closer to 49 than to 64, the square root is closer to 7. The difference from 49 is 6, and from 64 is 9. Step 3: Try 7.4: 7.4 × 7.4 = 54.76. This is 0.24 below 55. Step 4: Try 7.5: 7.5 × 7.5 = 56.25. This is 1.25 above 55. Step 5: Since 7.4^2 = 54.76 is only 0.24 below 55, and 7.5^2 = 56.25 is 1.25 above, 7.4 is the better approximation to the nearest tenth. The answer is 7.4.

8. √(48) + √(27) = ? Answer: 7√3 Solution: Simplify √(48) √(48) = √(16 × 3) = √(16) × √(3) = 4√3 Simplify √(27) √(27) = √(9 × 3) = √(9) × √(3) = 3√3 4√3 + 3√3 = (4 + 3)√3 = 7√3 The answer is 7√3.
   Full step-by-step solution

   Step 1: Simplify √(48) √(48) = √(16 × 3) = √(16) × √(3) = 4√3 Step 2: Simplify √(27) √(27) = √(9 × 3) = √(9) × √(3) = 3√3 Step 3: Add the simplified terms 4√3 + 3√3 = (4 + 3)√3 = 7√3 The answer is 7√3.