Order Numbers

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Answer Key & Explanations

Order Numbers · Grade 8 · Worksheet 1

1. Order: √(50), 7.1, 22/3, π² Answer: π² < 7.1 < √(50) < 22/3 Solution: Approximate each number as a decimal. - √(50) = √(25 × 2) = 5√(2) ≈ 5 × 1.414 = 7.07 - 7.1 is already a decimal. - 22/3 ≈ 7.333...
   Full step-by-step solution

   Step 1: Approximate each number as a decimal. - √(50) = √(25 × 2) = 5√(2) ≈ 5 × 1.414 = 7.07 - 7.1 is already a decimal. - 22/3 ≈ 7.333... - π² ≈ (3.14159)² ≈ 9.87 (but wait, that seems too large; let's recalculate: π ≈ 3.14, so π² ≈ 9.86. However, note that 9.86 is larger than all others, so the order might be different. Let's check carefully: Actually π² ≈ 9.87, which is greater than 7.33. So the order from least to greatest is: 7.07, 7.1, 7.33, 9.87. That would be √(50) < 7.1 < 22/3 < π². But wait, 7.07 is less than 7.1, so √(50) comes first. Let's re-evaluate: √(50) ≈ 7.071, 7.1 = 7.100, 22/3 ≈ 7.333, π² ≈ 9.870. So the correct order from least to greatest is: √(50), 7.1, 22/3, π². But the problem asks to order them, so we list them in increasing order: √(50) < 7.1 < 22/3 < π². Step 2: Write the final ordered list. The answer is √(50) < 7.1 < 22/3 < π².

2. Isabella is building a model of a solar system for her science class. She needs to place two planets on a number line representing distance from the Sun in astronomical units (AU). Planet X is located at √50 AU, and Planet Y is located at 7.2 AU. Isabella's friend Mason claims that Planet X is farther from the Sun than Planet Y. Is Mason correct? Explain your reasoning by comparing the two values. Answer: No, Mason is not correct. Planet X (√50 ≈ 7.07) is less than Planet Y (7.2), so Planet Y is farther from the Sun. Solution: Estimate √50. The perfect squares near 50 are 49 (7²) and 64 (8²). Since 50 is closer to 49, √50 is slightly more than 7.
   Full step-by-step solution

   Step 1: Estimate √50. The perfect squares near 50 are 49 (7²) and 64 (8²). Since 50 is closer to 49, √50 is slightly more than 7. √49 = 7, and 7.1² = 50.41, so √50 is approximately 7.07. Step 2: Compare √50 to 7.2. Since 7.07 < 7.2, √50 < 7.2. Step 3: Conclusion. Planet X is at √50 ≈ 7.07 AU, and Planet Y is at 7.2 AU. Since 7.07 < 7.2, Planet Y is farther from the Sun. Mason is not correct.

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 triangle's hypotenuse. What is the area of this circle? (Use π = 3.14) Answer: 78.5 Solution: Identify the triangle's hypotenuse. The triangle's 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 triangle's hypotenuse. The triangle's 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 Hypotenuse^2 = 36 + 64 Hypotenuse^2 = 100 Hypotenuse = sqrt(100) = 10 Step 3: Relate the hypotenuse to the circle. The problem says the circle's diameter is equal to the length of the hypotenuse. 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 of a circle = π * radius^2 Area = 3.14 * (5^2) Area = 3.14 * 25 Step 6: Perform the multiplication. 3.14 * 25 = 78.5 Final Answer: The area of the circle is 78.5.

4. Kaia is helping her family build a wooden fence. She has two options for the length of the fence panels. Option A uses panels that are each √65 feet long. Option B uses panels that are each 8.1 feet long. Kaia's father says the panels are almost the same length, but Kaia wants to know exactly which is longer. Which option has longer panels, and by approximately how much? Round your answer to the nearest hundredth. Answer: Option B is longer by approximately 0.04 feet. Solution: Estimate √65. The perfect squares near 65 are 64 (8^2) and 81 (9^2), so √65 is between 8 and 9. Since 65 is 1 more than 64, √65 is a little more than 8.
   Full step-by-step solution

   Step 1: Estimate √65. The perfect squares near 65 are 64 (8^2) and 81 (9^2), so √65 is between 8 and 9. Since 65 is 1 more than 64, √65 is a little more than 8. Using a calculator or approximation: √65 ≈ 8.06226. Step 2: Compare √65 and 8.1. Since 8.06226 < 8.1, Option B (8.1 feet) is longer. Step 3: Find the difference: 8.1 - 8.06226 = 0.03774. Step 4: Round to the nearest hundredth: 0.03774 rounds to 0.04. Answer: Option B is longer by approximately 0.04 feet.

5. Emma is comparing the thickness of two different types of wood for a shelf she is building. The first type of wood has a thickness of √45 millimeters. The second type of wood has a thickness of 6.5 millimeters. Which type of wood is thicker, and by approximately how many millimeters? Round your answer to the nearest tenth. Answer: The second type of wood (6.5 mm) is thicker by approximately 0.2 millimeters. Solution: Simplify √45. √45 = √(9 × 5) = 3√5. Step 2: Estimate the value of √5.
   Full step-by-step solution

   Step 1: Simplify √45. √45 = √(9 × 5) = 3√5. Step 2: Estimate the value of √5. Since 2^2 = 4 and 3^2 = 9, √5 is between 2 and 3, and closer to 2 because 5 is closer to 4. More precisely, √5 ≈ 2.236. Step 3: Multiply by 3: 3 × 2.236 = 6.708. So √45 ≈ 6.708 mm. Step 4: Compare to the second thickness, 6.5 mm. 6.708 > 6.5, so the first type (√45) is thicker. Step 5: Find the difference: 6.708 - 6.5 = 0.208 ≈ 0.2 mm (rounded to the nearest tenth). The answer is that the first type of wood (√45 mm) is thicker by about 0.2 millimeters.

6. Mason is helping his uncle build a rectangular wooden frame for a large mirror. The frame's length is √75 inches, and its width is 8 inches. His cousin Charlotte says the diagonal of the frame should be √128 inches, but Mason thinks it should be about 13.2 inches. Who is correct? Show your reasoning by comparing the exact diagonal length to 13.2. Answer: Mason is correct; the diagonal is √139 ≈ 11.8 inches, which is less than 13.2. Solution: Simplify the length. √75 = √(25 × 3) = 5√3. So length = 5√3 inches, width = 8 inches.
   Full step-by-step solution

   Step 1: Simplify the length. √75 = √(25 × 3) = 5√3. So length = 5√3 inches, width = 8 inches. Step 2: Use the Pythagorean theorem: diagonal² = (5√3)² + 8² = 25 × 3 + 64 = 75 + 64 = 139. Step 3: Diagonal = √139. Now estimate √139: 11² = 121, 12² = 144, so √139 is between 11 and 12, closer to 12 since 139 is closer to 144. More precisely, 11.8² = 139.24, 11.7² = 136.89, so √139 ≈ 11.8. Step 4: Compare √139 ≈ 11.8 to 13.2. Since 11.8 < 13.2, Mason is correct that the diagonal is not 13.2 inches; it is about 11.8 inches.

7. Mason is building a ramp for his skateboard. He needs to compare the lengths of three different ramp designs he sketched. Ramp A has a length of √72 inches. Ramp B has a length of 8.5 inches. Ramp C has a length of 17/2 inches. Mason wants to order the ramps from shortest to longest to decide which one to build. What is the correct order? Answer: Ramp C (8.5 inches), Ramp B (8.5 inches), Ramp A (approximately 8.485 inches) or Ramp A, then Ramp B and Ramp C (tied) Solution: Simplify √72. √72 = √(36×2) = 6√2. Now estimate √2 ≈ 1.414, so 6 × 1.414 = 8.484.
   Full step-by-step solution

   Step 1: Simplify √72. √72 = √(36×2) = 6√2. Now estimate √2 ≈ 1.414, so 6 × 1.414 = 8.484. So Ramp A is about 8.484 inches. Step 2: Ramp B is given as 8.5 inches. Step 3: Ramp C is 17/2 = 8.5 inches. Step 4: Compare: 8.484 < 8.5, so Ramp A is shortest. Ramp B and Ramp C are equal at 8.5 inches. Step 5: Order from shortest to longest: Ramp A (√72 inches ≈ 8.484 inches), then Ramp B and Ramp C (both 8.5 inches, tied). The correct order is Ramp A, then Ramp B and Ramp C.