Approximate Irrationals

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Approximate Irrationals · Grade 8 · Worksheet 2

1. Mason is building a square-shaped garden in his backyard. He wants the garden to have an area of 90 square feet so he can plant a variety of vegetables. To buy the right amount of fencing for the perimeter, he needs to know the side length of the garden. However, the fencing company only sells fence panels in whole foot lengths, so Mason needs to round the side length to the nearest whole number. What is the side length of Mason's garden in feet, rounded to the nearest whole number? Answer: 9 Solution: The side length of a square garden with area 90 square feet is sqrt(90) feet. Find two consecutive integers that sqrt(90) lies between.
   Full step-by-step solution

   Step 1: The side length of a square garden with area 90 square feet is sqrt(90) feet. Step 2: Find two consecutive integers that sqrt(90) lies between. Since 9^2 = 81 and 10^2 = 100, and 81 < 90 < 100, we know sqrt(90) is between 9 and 10. Step 3: Determine which integer is closer. Compare 90 to the midpoints: 90 - 81 = 9, and 100 - 90 = 10. Since 9 < 10, sqrt(90) is closer to 9 than to 10. Step 4: Therefore, rounding to the nearest whole number, the side length is 9 feet. The answer is 9.

2. A rectangular garden is drawn on a coordinate plane with corners at (0,0), (12,0), (12,5), and (0,5). A diagonal path is drawn from (0,0) to (12,5). What is the approximate length of this diagonal path? Round your answer to the nearest tenth. Answer: 13.0 Solution: The diagonal from (0,0) to (12,5) forms the hypotenuse of a right triangle with sides equal to the length and width of the rectangle. The horizontal side has length 12 - 0 = 12 units.
   Full step-by-step solution

   Step 1: The diagonal from (0,0) to (12,5) forms the hypotenuse of a right triangle with sides equal to the length and width of the rectangle. Step 2: The horizontal side has length 12 - 0 = 12 units. Step 3: The vertical side has length 5 - 0 = 5 units. Step 4: Use the Pythagorean theorem: diagonal^2 = 12^2 + 5^2 Step 5: Calculate: 12^2 = 144 and 5^2 = 25 Step 6: Add: 144 + 25 = 169 Step 7: Find the square root: sqrt(169) = 13 Step 8: Round to the nearest tenth: 13.0 The answer is 13.0.

3. Aroha is designing a square patio in her backyard. The patio has an area of 110 square feet. She wants to place a string of lights along the diagonal of the square from one corner to the opposite corner. What is the approximate length of the diagonal, rounded to the nearest tenth of a foot? Answer: 14.8 Solution: The area of the square is 110 square feet. So, side^2 = 110. The side length is sqrt(110).
   Full step-by-step solution

   Step 1: The area of the square is 110 square feet. So, side^2 = 110. The side length is sqrt(110). Step 2: sqrt(110) is between 10 and 11 because 10^2 = 100 and 11^2 = 121. Since 110 is closer to 121, sqrt(110) is approximately 10.5 (since 10.5^2 = 110.25). So side length is about 10.5 feet. Step 3: The diagonal of a square forms the hypotenuse of a right triangle with legs equal to the side length. By the Pythagorean theorem: diagonal^2 = side^2 + side^2 = 2 * side^2 = 2 * 110 = 220. Step 4: So diagonal = sqrt(220). sqrt(220) is between 14 and 15 because 14^2 = 196 and 15^2 = 225. 220 is closer to 225, so try 14.8: 14.8^2 = 219.04. Try 14.9: 14.9^2 = 222.01. Since 219.04 is closer to 220, sqrt(220) ≈ 14.8. Step 5: Rounding to the nearest tenth gives 14.8. The answer is 14.8.

4. √(18) + √(50) - √(8) = ? Answer: 6√2 Solution: Simplify √(18) = √(9×2) = √9 × √2 = 3√2 Simplify √(50) = √(25×2) = √25 × √2 = 5√2 Simplify √(8) = √(4×2) = √4 × √2 = 2√2 Substitute back: 3√2 + 5√2 - 2√2 Combine like terms: (3 + 5 - 2)√2 = 6√2 The answer is 6√2.
   Full step-by-step solution

   Step 1: Simplify √(18) = √(9×2) = √9 × √2 = 3√2 Step 2: Simplify √(50) = √(25×2) = √25 × √2 = 5√2 Step 3: Simplify √(8) = √(4×2) = √4 × √2 = 2√2 Step 4: Substitute back: 3√2 + 5√2 - 2√2 Step 5: Combine like terms: (3 + 5 - 2)√2 = 6√2 The answer is 6√2.

5. √(18) × √(2) = ? Answer: 6 Solution: Multiply the square roots: √(18) × √(2) = √(18 × 2) Multiply the numbers inside the radical: 18 × 2 = 36 Simplify the square root: √(36) = 6 The answer is 6.
   Full step-by-step solution

   Step 1: Multiply the square roots: √(18) × √(2) = √(18 × 2) Step 2: Multiply the numbers inside the radical: 18 × 2 = 36 Step 3: Simplify the square root: √(36) = 6 The answer is 6.

6. √(18) + √(8) = ? Answer: 5√2 Solution: Simplify √(18) √(18) = √(9 × 2) = √9 × √2 = 3√2 Simplify √(8) √(8) = √(4 × 2) = √4 × √2 = 2√2 3√2 + 2√2 = (3 + 2)√2 = 5√2 The answer is 5√2.
   Full step-by-step solution

   Step 1: Simplify √(18) √(18) = √(9 × 2) = √9 × √2 = 3√2 Step 2: Simplify √(8) √(8) = √(4 × 2) = √4 × √2 = 2√2 Step 3: Add the simplified expressions 3√2 + 2√2 = (3 + 2)√2 = 5√2 The answer is 5√2.

7. 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.

8. √(2) + √(8) = ? Answer: 3√2 Solution: Write the problem clearly. √(2) + √(8) Notice that 8 can be factored into 4 × 2. So √(8) = √(4 × 2) Use the property √(a × b) = √(a) × √(b).
   Full step-by-step solution

   Let's solve step by step. Step 1: Write the problem clearly. √(2) + √(8) Step 2: Notice that 8 can be factored into 4 × 2. So √(8) = √(4 × 2) Step 3: Use the property √(a × b) = √(a) × √(b). √(4 × 2) = √(4) × √(2) Step 4: Simplify √(4). √(4) = 2 So √(8) = 2 × √(2) Step 5: Substitute back into the original expression. √(2) + √(8) = √(2) + 2 × √(2) Step 6: Factor out √(2). √(2) × (1 + 2) = √(2) × 3 Step 7: Write the final answer. 3√(2) So the answer is 3√2.