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
Step 1: Set up the equation. Let the number be x. Its reciprocal is (1)/(x). The difference between the number and its reciprocal is 3.75. x - (1)/(x) = 3.75 Step 2: Convert the decimal to a fraction. 3.75 = (375)/(100) = (15)/(4) So the equation becomes: x - (1)/(x) = (15)/(4) Step 3: Clear the denominators by multiplying the entire equation by 4x. 4x (x - (1)/(x)) = 4x ((15)/(4)) 4x^2 - 4 = 15x Step 4: Rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. 4x^2 - 15x - 4 = 0 Step 5: Solve the quadratic equation by factoring. We look for two numbers that multiply to 4 × (-4) = -16 and add up to -15. These numbers are -16 and 1. 4x^2 - 16x + x - 4 = 0 Factor by grouping: 4x(x - 4) + 1(x - 4) = 0 (4x + 1)(x - 4) = 0 Step 6: Find the possible values for x. Set each factor to zero: 4x + 1 = 0 4x = -1 x = -(1)/(4) x - 4 = 0 x = 4 The number can be either 4 or -(1)/(4). The number is 4 or -(1)/(4). --- Step 1: Write down the given information and the formula for the area of a rectangle. Area (A) = 7462 cm^2 Width (W) = 82 cm Area formula: A = L × W We need to find the length (L). L = (A)/(W) = (7462)/(82) Step 2: Express the division using logarithms. L = ((7462)/(82)) L = (7462) - (82) Step 3: Find the logarithm of 7462 using a logarithm table. The characteristic for 7462 (a 4-digit number) is 4-1=3. From the logarithm table, the mantissa for 7462 (looking up 74, then 6, then mean difference for 2) is approximately 0.8727 + 0.0001 = 0.8728. (7462) = 3.8728 Step 4: Find the logarithm of 82 using a logarithm table. The characteristic for 82 (a 2-digit number) is 2-1=1. From the logarithm table, the mantissa for 82 (looking up 82, then 0) is approximately 0.9138. (82) = 1.9138 Step 5: Subtract the logarithms. L = 3.8728 - 1.9138 L = 1.9590 Step 6: Find the antilogarithm of 1.9590. The characteristic is 1, which means the number has 1+1=2 digits before the decimal point. The mantissa is 0.9590. From the antilogarithm table, the number corresponding to the mantissa 0.9590 is approximately 9.096. Therefore, L = 9.096 × 10^1. L = 90.96 cm The length of the farm is 90.96 cm. That's 2 down. 3 left today — send the next one.