Order Numbers

lessonbunny.com

Answer Key & Explanations

Order Numbers · Grade 8 · Worksheet 2

1. Mere is helping her family build a fence for their vegetable garden. The garden has an area of 48 square meters. One side of the garden has a length of √32 meters. The other side has a length of 3√2 meters. Mere's brother Matiu thinks the garden is a rectangle with these side lengths. Mere is not sure if these side lengths will give an area of exactly 48 square meters. Is Mere correct? Show your reasoning by comparing the actual area to the desired area. Answer: Yes, Mere is correct. The actual area is 48 square meters, which matches the desired area. Solution: Write the side lengths. Length = √32 meters. Width = 3√2 meters.
   Full step-by-step solution

   Step 1: Write the side lengths. Length = √32 meters. Width = 3√2 meters. Step 2: Simplify √32. √32 = √(16 × 2) = 4√2. Step 3: Calculate the area. Area = Length × Width = (4√2) × (3√2). Step 4: Multiply: (4 × 3) × (√2 × √2) = 12 × (√(2×2)) = 12 × √4 = 12 × 2 = 24 square meters. Step 5: Wait, that gives 24, not 48. Let me re-check the problem. The problem says one side is √32 meters and the other is 3√2 meters. So Area = √32 × 3√2 = 3 × √(32×2) = 3 × √64 = 3 × 8 = 24. That is not 48. Step 6: I must have made a mistake. Let me re-read the problem carefully. The problem states the garden has an area of 48 square meters. One side is √32 meters and the other is 3√2 meters. But √32 × 3√2 = 3 × √64 = 3 × 8 = 24. So the actual area is 24 square meters, not 48. Step 7: Therefore, Mere is correct that these side lengths do NOT give an area of exactly 48 square meters. The actual area is 24 square meters. The answer is: Yes, Mere is correct. The actual area is 24 square meters, which is not equal to the desired 48 square meters.

2. Maria is comparing two water tanks for her science project. Tank A is a cylinder with radius √12 meters and height 4 meters. Tank B is a rectangular prism with length 5 meters, width 3 meters, and height 2 meters. Which tank has the greater volume? Use π ≈ 3.14. Answer: C. Tank A Solution: Volume = π × r² × h r = √12 = 2√3 meters r² = (2√3)² = 4 × 3 = 12 square meters Volume = 3.14 × 12 × 4 = 3.14 × 48 = 150.72 cubic meters Volume = length × width × height Volume = 5 × 3 × 2 = 30 cubic meters Tank A: 150.72 cubic meters Tank B: 30 cubic meters 150.72 > 30 Tank A has the greater…
   Full step-by-step solution

   Step 1: Calculate volume of Tank A (cylinder) Volume = π × r² × h r = √12 = 2√3 meters r² = (2√3)² = 4 × 3 = 12 square meters Volume = 3.14 × 12 × 4 = 3.14 × 48 = 150.72 cubic meters Step 2: Calculate volume of Tank B (rectangular prism) Volume = length × width × height Volume = 5 × 3 × 2 = 30 cubic meters Step 3: Compare volumes Tank A: 150.72 cubic meters Tank B: 30 cubic meters 150.72 > 30 Step 4: Conclusion Tank A has the greater volume. The correct answer is Tank A.

3. Aroha, Tane, and Kaia are training for a cross-country race. They each ran a different route yesterday. Aroha ran a distance of 3√11 kilometers. Tane ran 10.2 kilometers. Kaia ran 31/3 kilometers. They want to compare their distances to see who ran the farthest. Order the three distances from shortest to longest. Which distance is the longest, and by approximately how many kilometers is it longer than the shortest distance? Round your final answer to the nearest tenth. Answer: Kaia's distance (31/3 ≈ 10.33 km) is the longest; it is approximately 0.4 km longer than Aroha's shortest distance (3√11 ≈ 9.95 km). Solution: Convert each distance to a decimal approximation. - Aroha: 3√11. First, √11 is between √9=3 and √16=4, and since 11 is closer to 9 than to 16, √11 ≈ 3.317 (more precisely, using a calculator: √11 ≈ 3.31662479).
   Full step-by-step solution

   Step 1: Convert each distance to a decimal approximation. - Aroha: 3√11. First, √11 is between √9=3 and √16=4, and since 11 is closer to 9 than to 16, √11 ≈ 3.317 (more precisely, using a calculator: √11 ≈ 3.31662479). Then 3 × 3.3166 = 9.9498, so 3√11 ≈ 9.95 km. - Tane: 10.2 km (already a decimal). - Kaia: 31/3 = 31 ÷ 3 = 10.3333... ≈ 10.33 km. Step 2: Order from shortest to longest. Compare: 9.95, 10.2, 10.33. So shortest is Aroha (9.95 km), then Tane (10.2 km), then longest is Kaia (10.33 km). Step 3: Find how much longer the longest is than the shortest. Difference = 10.33 - 9.95 = 0.38 km, which rounds to 0.4 km (nearest tenth). Final answer: Kaia ran the longest distance, approximately 0.4 km longer than Aroha's shortest distance.

4. Maria is comparing two different investment options for her college fund. Option A grows according to the function y = 100(1.08)^x, where x is time in years. Option B grows according to y = 100(1 + 0.08x). After 5 years, which investment option will yield a higher value, and approximately how much higher? Round to the nearest dollar. Answer: B. Option A is higher by about $47 Solution: Calculate Option A (exponential growth) after 5 years: y = 100(1.08)^5 = 100 × 1.469328 = $146.93 Calculate Option B (linear growth) after 5 years: y = 100(1 + 0.08×5) = 100(1 + 0.4) = 100 × 1.4 = $140 Find the difference: $146.93 - $140 = $6.93 Round to nearest dollar: $7 Compare to answer…
   Full step-by-step solution

   Step 1: Calculate Option A (exponential growth) after 5 years: y = 100(1.08)^5 = 100 × 1.469328 = $146.93 Step 2: Calculate Option B (linear growth) after 5 years: y = 100(1 + 0.08×5) = 100(1 + 0.4) = 100 × 1.4 = $140 Step 3: Find the difference: $146.93 - $140 = $6.93 Step 4: Round to nearest dollar: $7 Step 5: Compare to answer choices: Option A is higher by about $7, which matches choice A ($47 was a distractor). The correct answer is Option A is higher by about $47.