Order Numbers

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Answer Key & Explanations

Order Numbers · Grade 8 · Worksheet 3

1. Olivia is comparing two hiking trails. Trail A has a length of 5√5 miles, and Trail B has a length of 12 miles. She wants to know which trail is longer and by approximately how many miles. Order the two trail lengths on a number line to compare them. Round to the nearest tenth if needed. Answer: Trail B is longer by approximately 0.8 miles. Solution: Estimate the value of sqrt(5). Since 2^2 = 4 and 3^2 = 9, sqrt(5) is between 2 and 3. More precisely, sqrt(5) ≈ 2.236.
   Full step-by-step solution

   Step 1: Estimate the value of sqrt(5). Since 2^2 = 4 and 3^2 = 9, sqrt(5) is between 2 and 3. More precisely, sqrt(5) ≈ 2.236. Step 2: Calculate the length of Trail A: 5 × sqrt(5) ≈ 5 × 2.236 = 11.18 miles. Step 3: Trail B is 12 miles. Step 4: Compare: 11.18 < 12, so Trail B is longer. Step 5: Find the difference: 12 - 11.18 = 0.82 miles, which rounds to 0.8 miles. Step 6: On a number line, 11.18 (Trail A) is to the left of 12 (Trail B). The answer is: Trail B is longer by approximately 0.8 miles.

2. A rectangular prism is drawn with dimensions 8 cm by 12 cm by 15 cm. A diagonal is drawn from the bottom front left corner to the top back right corner. What is the exact length of this space diagonal? Answer: sqrt(433) Solution: The space diagonal of a rectangular prism can be found using the formula: d = sqrt(l² + w² + h²) Substitute the given dimensions: l = 8 cm, w = 12 cm, h = 15 cm Calculate each squared term: 8² = 64, 12² = 144, 15² = 225 Add the squared terms: 64 + 144 + 225 = 433 Take the square root: d =…
   Full step-by-step solution

   Step 1: The space diagonal of a rectangular prism can be found using the formula: d = sqrt(l² + w² + h²) Step 2: Substitute the given dimensions: l = 8 cm, w = 12 cm, h = 15 cm Step 3: Calculate each squared term: 8² = 64, 12² = 144, 15² = 225 Step 4: Add the squared terms: 64 + 144 + 225 = 433 Step 5: Take the square root: d = sqrt(433) The exact length of the space diagonal is sqrt(433) cm.

3. Liam is designing a rectangular garden with a length of √50 meters and a width of 3√2 meters. His friend Emma is designing a square garden with side length 7 meters. Which garden has the larger area, and by approximately how many square meters? Round your answer to the nearest tenth. Answer: Liam's garden is larger by approximately 0.9 square meters Solution: When comparing areas involving rational and irrational numbers, it's helpful to simplify square roots and calculate exact values before approximating. For rectangles, multiply length by width.
   Full step-by-step solution

   When comparing areas involving rational and irrational numbers, it's helpful to simplify square roots and calculate exact values before approximating. For rectangles, multiply length by width. For squares, square the side length. Rational numbers can be expressed as decimals for comparison, but maintaining exact values initially preserves accuracy.

4. Liam is designing a rectangular garden with a length of √50 meters and a width of 3√2 meters. His friend Emma is designing a square garden with a side length of 4√3 meters. Which garden has the larger area, and by approximately how many square meters? Round your answer to the nearest tenth. Answer: Liam's garden by 1.8 square meters Solution: When comparing areas involving irrational numbers, it's often helpful to simplify radical expressions first.
   Full step-by-step solution

   When comparing areas involving irrational numbers, it's often helpful to simplify radical expressions first. For example, √8 can be simplified to 2√2. To compare two irrational numbers, you can either simplify them to have the same radical term or convert them to decimal approximations. The area of a rectangle is found by multiplying length and width, while the area of a square is the side length squared.

5. Order: √(53), 7.3, 22/3, π² Answer: 7.3 < √(53) < 22/3 < π² Solution: Estimate √(53). Since 7² = 49 and 8² = 64, √(53) is between 7 and 8. 53 - 49 = 4, so √(53) ≈ 7.28.
   Full step-by-step solution

   Step 1: Estimate √(53). Since 7² = 49 and 8² = 64, √(53) is between 7 and 8. 53 - 49 = 4, so √(53) ≈ 7.28. Step 2: 7.3 is already a decimal. Step 3: 22/3 = 22 ÷ 3 = 7.333... = 7.3̅. Step 4: π² ≈ (3.1416)² = 9.8696. Step 5: Compare decimals: 7.3 < 7.28 < 7.333... < 9.8696. Step 6: Order from least to greatest: 7.3, √(53), 22/3, π². The answer is 7.3 < √(53) < 22/3 < π².

6. A scientist is comparing the sizes of two microscopic organisms. Organism A has a diameter of 3√8 micrometers, while Organism B has a diameter of 2√18 micrometers. Which organism is larger, or are they the same size? Answer: A. They are the same size Solution: Simplify 3√8 √8 = √(4×2) = 2√2 3√8 = 3 × 2√2 = 6√2 Simplify 2√18 √18 = √(9×2) = 3√2 2√18 = 2 × 3√2 = 6√2 Both organisms have a diameter of 6√2 micrometers Since both diameters are equal, the organisms are the same size.
   Full step-by-step solution

   Step 1: Simplify 3√8 √8 = √(4×2) = 2√2 3√8 = 3 × 2√2 = 6√2 Step 2: Simplify 2√18 √18 = √(9×2) = 3√2 2√18 = 2 × 3√2 = 6√2 Step 3: Compare the simplified expressions Both organisms have a diameter of 6√2 micrometers Step 4: Conclusion Since both diameters are equal, the organisms are the same size. The correct answer is They are the same size.