Order Numbers

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Answer Key & Explanations

Order Numbers · Grade 8 · Worksheet 3

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is inscribed inside this triangle, tangent to all three sides. What is the approximate area of the circle? (Use π ≈ 3.14) Answer: 12.56 Solution: The triangle has vertices at (0,0), (6,0), and (6,8). This is a right triangle with legs along the x-axis and a vertical line. Leg lengths: horizontal leg = 6, vertical leg = 8.
   Full step-by-step solution

   Step 1: Understand the triangle and circle The triangle has vertices at (0,0), (6,0), and (6,8). This is a right triangle with legs along the x-axis and a vertical line. Leg lengths: horizontal leg = 6, vertical leg = 8. Step 2: Find the hypotenuse Hypotenuse length = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10. Step 3: Inradius formula for a right triangle For a right triangle, the inradius r = (leg1 + leg2 - hypotenuse) / 2. So r = (6 + 8 - 10) / 2 = (4) / 2 = 2. Step 4: Area of the circle Area = π * r^2 = π * (2^2) = 4π. Using π ≈ 3.14, area ≈ 4 * 3.14 = 12.56. Step 5: Conclusion The approximate area of the inscribed circle is 12.56.

2. Liam is designing a rectangular garden with an area of 72 square meters. He wants the length to be the square root of 128 meters and the width to be 4 times the square root of 2 meters. His friend Noah says the dimensions won't work because the area won't match. Help Liam determine if his planned dimensions will give him exactly 72 square meters of garden area. Answer: B. yes Solution: When multiplying expressions containing square roots, you can multiply the numbers outside the square roots together and the numbers inside the square roots together.
   Full step-by-step solution

   When multiplying expressions containing square roots, you can multiply the numbers outside the square roots together and the numbers inside the square roots together. Then simplify the resulting square root by factoring out perfect squares. This process helps verify if the product of two irrational numbers results in a rational number.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (12,5). A circle is drawn such that its diameter is equal to the length of the triangle's hypotenuse. What is the exact area of this circle in terms of π? Answer: 169π/4 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 12 units and 5 units.
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs of the triangle are 12 units and 5 units. Hypotenuse = sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13 units Step 2: The circle's diameter equals the hypotenuse length, so diameter = 13 units. Step 3: Find the radius of the circle. Radius = diameter/2 = 13/2 units Step 4: Calculate the area of the circle. Area = π × (radius)² = π × (13/2)² = π × (169/4) = 169π/4 The exact area of the circle is 169π/4.

4. Aisha is comparing two water tanks for her science project. Tank A is a cylinder with radius √12 meters and height 5 meters. Tank B is a rectangular prism with length 4 meters, width 3 meters, and height 6 meters. Which tank has the greater volume? Use π ≈ 3.14 and round to the nearest tenth if needed. Answer: C. Tank A Solution: Volume = π × radius² × height Radius = √12 meters Radius² = (√12)² = 12 square meters Volume = 3.14 × 12 × 5 = 3.14 × 60 = 188.4 cubic meters Volume = length × width × height Volume = 4 × 3 × 6 = 12 × 6 = 72 cubic meters Tank A volume = 188.4 cubic meters Tank B volume = 72 cubic meters 188.4 >…
   Full step-by-step solution

   Step 1: Calculate volume of Tank A (cylinder) Volume = π × radius² × height Radius = √12 meters Radius² = (√12)² = 12 square meters Volume = 3.14 × 12 × 5 = 3.14 × 60 = 188.4 cubic meters Step 2: Calculate volume of Tank B (rectangular prism) Volume = length × width × height Volume = 4 × 3 × 6 = 12 × 6 = 72 cubic meters Step 3: Compare the volumes Tank A volume = 188.4 cubic meters Tank B volume = 72 cubic meters 188.4 > 72 Step 4: Conclusion Tank A has the greater volume. The correct answer is Tank A.

5. Mason is building a wooden frame for a rectangular art display. The length of the frame must be √72 inches, and the width must be 2√2 inches. His friend Charlotte suggests that the area of the frame is exactly 27 square inches. Is Charlotte correct? Justify your answer by comparing the exact area to 27. Answer: No, the area is 24 square inches, not 27. Solution: Simplify the length: √72 = √(36 × 2) = 6√2. The width is already 2√2. Multiply length and width to find area: (6√2) × (2√2) = 6 × 2 × √2 × √2 = 12 × 2 = 24.
   Full step-by-step solution

   Step 1: Simplify the length: √72 = √(36 × 2) = 6√2. Step 2: The width is already 2√2. Step 3: Multiply length and width to find area: (6√2) × (2√2) = 6 × 2 × √2 × √2 = 12 × 2 = 24. Step 4: The exact area is 24 square inches. Step 5: Compare to 27: 24 ≠ 27, so Charlotte is incorrect. The answer is No, the area is 24 square inches, not 27.

6. √(2.25 × 10²) = ? Answer: 15 Solution: We have √(2.25 × 10²). This means: square root of (2.25 multiplied by 10 squared). Compute 10² 10² = 10 × 10 = 100.
   Full step-by-step solution

   Let's solve step by step. Step 1: Understand the expression We have √(2.25 × 10²). This means: square root of (2.25 multiplied by 10 squared). Step 2: Compute 10² 10² = 10 × 10 = 100. Step 3: Multiply 2.25 by 100 2.25 × 100 = 225. So the expression becomes √225. Step 4: Find the square root of 225 We ask: which number multiplied by itself gives 225? 15 × 15 = 225. So √225 = 15. Step 5: Final answer Thus, √(2.25 × 10²) = 15.