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: Express g(x) in the form R(2x+). We are given g(x) = 5(2x) + 12(2x). We want to express this in the form R(2x+), where R > 0 and is an acute angle. Using the compound angle formula, R(2x+) = R((2x) + (2x)). R(2x+) = (R)(2x) + (R)(2x) Comparing this to 5(2x) + 12(2x), we have: R = 5 (1) R = 12 (2) To find R, square and add (1) and (2): (R)^2 + (R)^2 = 5^2 + 12^2 R^2(^2 + ^2) = 25 + 144 Since ^2 + ^2 = 1: R^2 = 169 Since R > 0: R = sqrt(169) = 13 To find , divide (2) by (1): (R)/(R) = (12)/(5) = (12)/(5) Since is an acute angle, we find its value: = ((12)/(5)) ≈ 67.38^ Therefore, g(x) can be expressed as: g(x) = 13(2x + ((12)/(5))) or approximately g(x) = 13(2x + 67.38^) Step 2: Find the minimum value of (26)/(g(x)) + 15. Substitute the expression for g(x) into the given expression: (26)/(g(x)) + 15 = (26)/(13(2x+)) + 15 = (2)/((2x+)) + 15 The range of the sine function is -1 (2x+) 1. For the expression (2)/((2x+)) + 15 to be minimum, the term (2)/((2x+)) must be minimum. This occurs when (2x+) is negative and has the largest possible magnitude, which is (2x+) = -1. Substitute (2x+) = -1 into the expression: Minimum value = (2)/(-1) + 15 = -2 + 15 = 13 The minimum value of (26)/(g(x)) + 15 is 13. 3 done, 2 left today. You're making progress.