This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the calculations using mathematical tables: i) Calculate (364.6)^3 Let X = (364.6)^3. Step 1: Find the logarithm of 364.6. The characteristic of (364.6) is 2 (since there are 3 digits before the decimal point, 3-1=2). From the logarithm table, the mantissa for 364 is 5611. The mean difference for 6 is 7. So, the mantissa for 364.6 is 5611 + 7 = 5618. (364.6) = 2.5618 Step 2: Multiply the logarithm by 3. X = 3 × (364.6) X = 3 × 2.5618 X = 7.6854 Step 3: Find the antilogarithm of 7.6854. The characteristic is 7, so the number will have 7+1=8 digits before the decimal point. From the antilogarithm table, the antilog of 0.685 is 4.842. The mean difference for 4 is 5. So, the antilog of 0.6854 is 4.842 + 0.005 = 4.847. X = 4.847 × 10^7 X = 48,470,000 The value is 48,470,000. ii) Calculate (0.6492)^2 Let Y = (0.6492)^2. Step 1: Find the logarithm of 0.6492. The characteristic of (0.6492) is 1 (since there is one zero immediately after the decimal point, 0-1=-1). From the logarithm table, the mantissa for 649 is 8122. The mean difference for 2 is 1. So, the mantissa for 0.6492 is 8122 + 1 = 8123. (0.6492) = 1.8123 Step 2: Multiply the logarithm by 2. Y = 2 × (0.6492) Y = 2 × (1 + 0.8123) Y = -2 + 1.6246 Y = -1 + 0.6246 Y = 1.6246 Step 3: Find the antilogarithm of 1.6246. The characteristic is 1, so the number will have one zero before the first non-zero digit. From the antilogarithm table, the antilog of 0.624 is 4.207. The mean difference for 6 is 6. So, the antilog of 0.6246 is 4.207 + 0.006 = 4.213. Y = 0.4213 The value is 0.4213. That's 2 down. 3 left today — send the next one.