Rational vs Irrational

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Answer Key & Explanations

Rational vs Irrational · Grade 8 · Worksheet 1

1. Aroha is building a square patio in her backyard. The area of the patio is 45 square meters. She needs to know if the side length of the patio is a rational or irrational number so she can order the correct number of paving stones. What type of number is the side length of Aroha's patio? Answer: irrational Solution: The area of a square is given by A = s^2, where s is the side length. The area is 45 square meters, so s^2 = 45. Take the square root of both sides: s = sqrt(45).
   Full step-by-step solution

   Step 1: The area of a square is given by A = s^2, where s is the side length. Step 2: The area is 45 square meters, so s^2 = 45. Step 3: Take the square root of both sides: s = sqrt(45). Step 4: Simplify sqrt(45) by factoring: sqrt(45) = sqrt(9 * 5) = sqrt(9) * sqrt(5) = 3 * sqrt(5). Step 5: sqrt(5) is irrational because 5 is not a perfect square and its square root cannot be written as a fraction. Multiplying an irrational number (sqrt(5)) by a rational number (3) still gives an irrational number. Step 6: Therefore, the side length of Aroha's patio is an irrational number.

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), and a circular fountain is placed at the exact midpoint of this diagonal. What is the distance from the fountain to the corner at (12,0)? Round your answer to the nearest tenth. Answer: 6.5 Solution: Find the midpoint of the diagonal from (0,0) to (12,5) Midpoint formula: ((x1+x2)/2, (y1+y2)/2) Midpoint = ((0+12)/2, (0+5)/2) = (12/2, 5/2) = (6, 2.5) Calculate the distance from the fountain at (6,2.5) to the corner at (12,0) Distance formula: sqrt((x2-x1)^2 + (y2-y1)^2) Distance =…
   Full step-by-step solution

   Step 1: Find the midpoint of the diagonal from (0,0) to (12,5) Midpoint formula: ((x1+x2)/2, (y1+y2)/2) Midpoint = ((0+12)/2, (0+5)/2) = (12/2, 5/2) = (6, 2.5) Step 2: Calculate the distance from the fountain at (6,2.5) to the corner at (12,0) Distance formula: sqrt((x2-x1)^2 + (y2-y1)^2) Distance = sqrt((12-6)^2 + (0-2.5)^2) = sqrt(6^2 + (-2.5)^2) = sqrt(36 + 6.25) = sqrt(42.25) Step 3: Simplify the square root sqrt(42.25) = 6.5 The answer is 6.5.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is drawn such that its diameter is equal to the length of the hypotenuse of this triangle. What is the area of the circle? (Use π = 3.14) Answer: 78.5 Solution: Identify the hypotenuse of the triangle. The triangle vertices are (0,0), (6,0), and (6,8). The side from (0,0) to (6,0) is horizontal, length = 6.
   Full step-by-step solution

   Step 1: Identify the hypotenuse of the triangle. The triangle vertices are (0,0), (6,0), and (6,8). The side from (0,0) to (6,0) is horizontal, length = 6. The side from (6,0) to (6,8) is vertical, length = 8. The hypotenuse is from (0,0) to (6,8). Step 2: Calculate the hypotenuse length using the Pythagorean theorem. Hypotenuse^2 = 6^2 + 8^2 = 36 + 64 = 100. Hypotenuse = square root of 100 = 10. Step 3: Relate the hypotenuse to the circle. The problem says the circle's diameter equals the hypotenuse length. So diameter of circle = 10. Step 4: Find the radius of the circle. Radius = diameter / 2 = 10 / 2 = 5. Step 5: Calculate the area of the circle. Area = π × radius^2 = 3.14 × 5^2 = 3.14 × 25. Step 6: Multiply. 3.14 × 25 = 78.5. Final answer: 78.5

4. Ava is building a square garden in her backyard. She wants the garden to have an area of 81 square feet. She tells her friend Noah that the side length of the garden must be a rational number. Is Ava correct? Explain why or why not. Answer: Yes, Ava is correct because the square root of 81 is 9, which is a rational number. Solution: Recall the formula for the area of a square: A = s^2, where s is the side length. Step 2: We know the area is 81 square feet, so s^2 = 81. Step 3: Take the square root of both sides: s = sqrt(81).
   Full step-by-step solution

   Step 1: Recall the formula for the area of a square: A = s^2, where s is the side length. Step 2: We know the area is 81 square feet, so s^2 = 81. Step 3: Take the square root of both sides: s = sqrt(81). Step 4: sqrt(81) = 9, because 9 * 9 = 81. Step 5: A rational number is any number that can be expressed as a fraction a/b where a and b are integers and b is not zero. The number 9 can be written as 9/1, so it is rational. Step 6: Therefore, Ava is correct: the side length is 9, which is a rational number.

5. Is √(121) + ∛(343) rational or irrational? Explain. Answer: Rational Solution: Evaluate √(121). Since 11 × 11 = 121, √(121) = 11. Evaluate ∛(343).
   Full step-by-step solution

   Step 1: Evaluate √(121). Since 11 × 11 = 121, √(121) = 11. Step 2: Evaluate ∛(343). Since 7 × 7 × 7 = 343, ∛(343) = 7. Step 3: Add the results: 11 + 7 = 18. Step 4: 18 is an integer. Every integer is a rational number because it can be written as a fraction (18/1). Therefore, √(121) + ∛(343) is rational.

6. Hana is helping her younger brother understand rational and irrational numbers. She writes down two numbers on a piece of paper: sqrt(121) and sqrt(37). She asks her brother to classify each one as rational or irrational. What are the correct classifications for sqrt(121) and sqrt(37)? Answer: sqrt(121) is rational, sqrt(37) is irrational Solution: Look at sqrt(121). Ask: Is 121 a perfect square? 121 = 11 × 11, so sqrt(121) = 11.
   Full step-by-step solution

   Step 1: Look at sqrt(121). Ask: Is 121 a perfect square? 121 = 11 × 11, so sqrt(121) = 11. Since 11 can be written as 11/1, it is a rational number. Step 2: Look at sqrt(37). Ask: Is 37 a perfect square? The perfect squares near 37 are 36 (6²) and 49 (7²). 37 is not a perfect square, so sqrt(37) cannot be expressed as a fraction of two integers. It is an irrational number. Step 3: Conclusion: sqrt(121) is rational, and sqrt(37) is irrational.

7. Is √(361) rational or irrational? Answer: Rational Solution: Determine if 361 is a perfect square. 19 × 19 = 361, so 361 is a perfect square. Step 2: √(361) = 19.
   Full step-by-step solution

   Step 1: Determine if 361 is a perfect square. 19 × 19 = 361, so 361 is a perfect square. Step 2: √(361) = 19. Step 3: 19 is a whole number. Any whole number can be written as a fraction (19/1), so it is rational. The answer is Rational.

8. Isabella is helping her science teacher classify numbers on the number line. She has two numbers: the square root of 27 and the fraction 27/7. She needs to determine whether each number is rational or irrational and explain her reasoning. What is the classification for each number? Answer: sqrt(27) is irrational, 27/7 is rational Solution: Consider sqrt(27). 27 is not a perfect square (the perfect squares near it are 25 and 36). Simplify sqrt(27) = sqrt(9 * 3) = 3 * sqrt(3).
   Full step-by-step solution

   Step 1: Consider sqrt(27). 27 is not a perfect square (the perfect squares near it are 25 and 36). Simplify sqrt(27) = sqrt(9 * 3) = 3 * sqrt(3). Since sqrt(3) is irrational (cannot be written as a fraction), and multiplying an irrational number by a rational number (3) gives an irrational number, sqrt(27) is irrational. Step 2: Consider 27/7. This is already written as a fraction of two integers (27 and 7). Any number that can be expressed as a fraction of two integers is rational. Therefore, 27/7 is rational. Final answer: sqrt(27) is irrational; 27/7 is rational.