Approximate Irrationals

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Answer Key & Explanations

Approximate Irrationals · Grade 8 · Worksheet 3

1. √(18) × √(2) = ? Answer: 6 Solution: Multiply the square roots: √(18) × √(2) = √(18 × 2) Multiply the numbers inside the radical: 18 × 2 = 36 Simplify the square root: √(36) = 6 The answer is 6.
   Full step-by-step solution

   Step 1: Multiply the square roots: √(18) × √(2) = √(18 × 2) Step 2: Multiply the numbers inside the radical: 18 × 2 = 36 Step 3: Simplify the square root: √(36) = 6 The answer is 6.

2. Hana is painting a mural on a square wall. The area of the wall is 128 square feet. She needs to estimate the side length of the wall to the nearest tenth of a foot so she can buy the right amount of paint tape. What is the approximate side length of the wall? Answer: 11.3 feet Solution: The side length of a square is the square root of its area. So side length = sqrt(128). Find the two perfect squares that 128 is between.
   Full step-by-step solution

   Step 1: The side length of a square is the square root of its area. So side length = sqrt(128). Step 2: Find the two perfect squares that 128 is between. 11^2 = 121 and 12^2 = 144. Since 128 is between 121 and 144, sqrt(128) is between 11 and 12. Step 3: Since 128 is closer to 121 than to 144, try 11.3. 11.3^2 = 127.69. This is slightly less than 128. Step 4: Try 11.4. 11.4^2 = 129.96. This is greater than 128. Step 5: Since 127.69 is closer to 128 than 129.96 is, the best estimate to the nearest tenth is 11.3. The approximate side length of the wall is 11.3 feet.

3. Hana is building a square-shaped garden in her backyard. She wants the garden to have an area of 98 square feet so she can plant a variety of flowers. She needs to buy fencing to go around the entire garden, but the fencing is sold only in whole foot lengths. To the nearest foot, how many feet of fencing should Hana buy? Answer: 40 Solution: The area of the square garden is 98 square feet. The side length s = sqrt(98). Find the perfect squares closest to 98: 9^2 = 81 and 10^2 = 100.
   Full step-by-step solution

   Step 1: The area of the square garden is 98 square feet. The side length s = sqrt(98). Step 2: Find the perfect squares closest to 98: 9^2 = 81 and 10^2 = 100. Since 98 is closer to 100 than to 81, sqrt(98) is approximately 9.9 (since 9.9^2 = 98.01, which is very close to 98). Step 3: The perimeter of a square is 4 times the side length: P = 4 * 9.9 = 39.6 feet. Step 4: Round 39.6 to the nearest whole foot: 40 feet. The answer is 40.

4. Liam is designing a circular garden with a diameter of 12 meters. He needs to calculate the circumference to purchase the right amount of border fencing. Using π ≈ 3.14, what is the approximate circumference of Liam's garden? Answer: 37.68 Solution: We are given the diameter of a circular garden as 12 meters. We need the circumference, which is the distance around the circle. C = π × d where d is the diameter and π is approximately 3.14.
   Full step-by-step solution

   Step 1: Understand the problem We are given the diameter of a circular garden as 12 meters. We need the circumference, which is the distance around the circle. Step 2: Recall the formula for circumference The circumference C of a circle is given by: C = π × d where d is the diameter and π is approximately 3.14. Step 3: Substitute the known values into the formula Diameter d = 12 meters π ≈ 3.14 So: C = 3.14 × 12 Step 4: Perform the multiplication First, multiply 3.14 by 10: 3.14 × 10 = 31.4 Then multiply 3.14 by 2: 3.14 × 2 = 6.28 Now add them: 31.4 + 6.28 = 37.68 Alternatively, you can multiply directly: 3.14 × 12 = 37.68 Step 5: State the final answer The approximate circumference of Liam's garden is 37.68 meters.

5. A scientist is studying bacteria growth. The number of bacteria after t hours is given by the function N(t) = 2^t. Between which two consecutive whole numbers does t lie when N(t) = 150? Answer: 7 Solution: We need to find t such that 2^t = 150 Calculate powers of 2 around 150: 2^7 = 128 2^8 = 256 Since 128 < 150 < 256, and 2^t is an increasing function Therefore, t must be between 7 and 8 The question asks for the lower of the two consecutive whole numbers The answer is 7
   Full step-by-step solution

   Step 1: We need to find t such that 2^t = 150 Step 2: Calculate powers of 2 around 150: 2^7 = 128 2^8 = 256 Step 3: Since 128 < 150 < 256, and 2^t is an increasing function Step 4: Therefore, t must be between 7 and 8 Step 5: The question asks for the lower of the two consecutive whole numbers Step 6: The answer is 7

6. √(45) ≈ ? (to the nearest tenth) Answer: 6.7 Solution: Identify perfect squares near 45. 6^2 = 36 and 7^2 = 49. So √45 is between 6 and 7.
   Full step-by-step solution

   Step 1: Identify perfect squares near 45. 6^2 = 36 and 7^2 = 49. So √45 is between 6 and 7. Step 2: Since 45 is closer to 49 than to 36, the square root is closer to 7. The difference from 49 is 4, and from 36 is 9. Step 3: Try 6.7: 6.7 × 6.7 = 44.89. This is very close to 45. Step 4: Try 6.8: 6.8 × 6.8 = 46.24. This is farther from 45. Step 5: Since 6.7^2 = 44.89 is only 0.11 below 45, and 6.8^2 = 46.24 is 1.24 above, 6.7 is the better approximation to the nearest tenth. The answer is 6.7.

7. √(86) ≈ ? (to the nearest tenth) Answer: 9.3 Solution: Identify perfect squares near 86. 9^2 = 81 and 10^2 = 100. So √86 is between 9 and 10.
   Full step-by-step solution

   Step 1: Identify perfect squares near 86. 9^2 = 81 and 10^2 = 100. So √86 is between 9 and 10. Step 2: Since 86 is closer to 81 than to 100, the square root is closer to 9. The difference from 81 is 5, and from 100 is 14. Step 3: Try 9.3: 9.3 × 9.3 = 86.49. This is 0.49 above 86. Step 4: Try 9.2: 9.2 × 9.2 = 84.64. This is 1.36 below 86. Step 5: Since 9.3^2 = 86.49 is only 0.49 above 86, and 9.2^2 = 84.64 is 1.36 below, 9.3 is the better approximation to the nearest tenth. The answer is 9.3.