Rational vs Irrational

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Answer Key & Explanations

Rational vs Irrational · Grade 8 · Worksheet 2

1. Charlotte draws a square on a coordinate plane with vertices at (0,0), (9,0), (9,9), and (0,9). She then draws a diagonal from (0,0) to (9,9). At the midpoint of this diagonal, she places a small circle. A second circle is drawn with its center at (9,0) and a radius equal to the distance from (9,0) to the midpoint of the diagonal. Determine whether the radius of the second circle is rational or irrational, and explain why. Answer: irrational Solution: Find the midpoint of the diagonal from (0,0) to (9,9). Midpoint = ((0+9)/2, (0+9)/2) = (9/2, 9/2) = (4.5, 4.5) Find the distance from (9,0) to (4.5, 4.5) using the distance formula.
   Full step-by-step solution

   Step 1: Find the midpoint of the diagonal from (0,0) to (9,9). Midpoint = ((0+9)/2, (0+9)/2) = (9/2, 9/2) = (4.5, 4.5) Step 2: Find the distance from (9,0) to (4.5, 4.5) using the distance formula. Distance = sqrt((4.5 - 9)^2 + (4.5 - 0)^2) Distance = sqrt((-4.5)^2 + (4.5)^2) Distance = sqrt(20.25 + 20.25) Distance = sqrt(40.5) Step 3: Simplify sqrt(40.5). 40.5 = 81/2 sqrt(40.5) = sqrt(81/2) = sqrt(81)/sqrt(2) = 9/sqrt(2) = (9*sqrt(2))/2 Step 4: Determine if this is rational or irrational. sqrt(2) is irrational. Multiplying an irrational number by a rational number (9/2) still gives an irrational number. Therefore, the radius is irrational. The answer is irrational.

2. Is √(196) + ∛(216) rational or irrational? Answer: Rational Solution: Evaluate √(196). Since 14 × 14 = 196, √(196) = 14. Evaluate ∛(216).
   Full step-by-step solution

   Step 1: Evaluate √(196). Since 14 × 14 = 196, √(196) = 14. Step 2: Evaluate ∛(216). Since 6 × 6 × 6 = 216, ∛(216) = 6. Step 3: Add the results: 14 + 6 = 20. Step 4: 20 can be written as 20/1, which is a fraction of two integers. Therefore, 20 is a rational number. The answer is Rational.

3. Is √(121) + ∛(216) rational or irrational? Explain. Answer: Rational Solution: Evaluate √(121). Since 11 × 11 = 121, √(121) = 11. Evaluate ∛(216).
   Full step-by-step solution

   Step 1: Evaluate √(121). Since 11 × 11 = 121, √(121) = 11. Step 2: Evaluate ∛(216). Since 6 × 6 × 6 = 216, ∛(216) = 6. Step 3: Add the results: 11 + 6 = 17. Step 4: 17 is an integer, and all integers are rational numbers because they can be written as a fraction (17/1). Therefore, √(121) + ∛(216) is rational.

4. Emma is helping her science class calculate the dimensions of a rectangular terrarium for a project. The area of the terrarium must be exactly 125 square inches. The teacher says the length should be the square root of 75 inches, and the width should be the square root of 75 inches as well. Emma's friend Liam claims the side length of the terrarium will be a rational number because 75 ends in 5. Is Liam correct? Determine whether the side length is rational or irrational, and explain your reasoning. Answer: Irrational Solution: The side length is sqrt(75) inches. Step 2: Determine if 75 is a perfect square. The perfect squares near 75 are 64 (8^2) and 81 (9^2).
   Full step-by-step solution

   Step 1: The side length is sqrt(75) inches. Step 2: Determine if 75 is a perfect square. The perfect squares near 75 are 64 (8^2) and 81 (9^2). Since 75 is not a perfect square, sqrt(75) cannot be written as an integer. Step 3: Simplify sqrt(75): sqrt(75) = sqrt(25 * 3) = sqrt(25) * sqrt(3) = 5 * sqrt(3). Step 4: sqrt(3) is irrational because it cannot be expressed as a fraction of two integers and its decimal representation does not terminate or repeat. Step 5: Multiplying a rational number (5) by an irrational number (sqrt(3)) gives an irrational number. Step 6: Therefore, sqrt(75) is irrational. Liam is incorrect; the side length is irrational.

5. 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 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 vertices are (0,0), (6,0), and (6,8).
   Full step-by-step solution

   Step 1: Identify the hypotenuse of the triangle. The vertices are (0,0), (6,0), and (6,8). The points (0,0) and (6,8) are the endpoints of the hypotenuse because the other two sides are along the axes: - From (0,0) to (6,0) is horizontal length 6 - From (6,0) to (6,8) is vertical length 8 Step 2: Calculate the length of the hypotenuse using the Pythagorean theorem. Hypotenuse length = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 Step 3: Relate the hypotenuse to the circle. The hypotenuse is the diameter of the circle. So, diameter d = 10 Step 4: Find the radius. Radius r = d / 2 = 10 / 2 = 5 Step 5: Calculate the area of the circle. Area = π * r^2 = 3.14 * (5^2) = 3.14 * 25 Step 6: Multiply. 3.14 * 25 = 78.5 Final answer: 78.5

6. √(2 + 2) + ∛8 = ? Answer: 4 Solution: We have: √(2 + 2) + ∛8 Simplify inside the square root first. 2 + 2 = 4 So √(2 + 2) = √4 Evaluate √4. √4 means the positive square root of 4, which is 2.
   Full step-by-step solution

   Let's solve step-by-step. We have: √(2 + 2) + ∛8 Step 1: Simplify inside the square root first. 2 + 2 = 4 So √(2 + 2) = √4 Step 2: Evaluate √4. √4 means the positive square root of 4, which is 2. So √(2 + 2) = 2. Step 3: Evaluate ∛8. ∛8 means the cube root of 8. Since 2 × 2 × 2 = 8, ∛8 = 2. Step 4: Add the results. 2 + 2 = 4. Final answer: 4

7. Liam is designing a rectangular garden with an area of 24 square meters. He wants the length to be √18 meters and the width to be √8 meters. His friend Noah says the area won't be exactly 24 square meters because these are irrational numbers. Is Noah correct? Explain why or why not. Answer: No, Noah is not correct. The area will be exactly 24 square meters because √18 × √8 = √144 = 12, and 12 × 2 = 24. Solution: When working with square roots, multiplying two irrational numbers can sometimes produce a rational result if the product under the radical is a perfect square.
   Full step-by-step solution

   When working with square roots, multiplying two irrational numbers can sometimes produce a rational result if the product under the radical is a perfect square. For example, √2 × √8 = √16 = 4, which is rational. This occurs because the irrational parts can cancel out or combine to form rational numbers.