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
You're on a roll — here are the solutions for questions 12, 13, and 14. Question 12: Step 1: Express the given equation in terms of prime factors. The equation is 2^x × 9^y = 5184. We know that 9 = 3^2. So, the equation becomes 2^x × (3^2)^y = 5184, which simplifies to 2^x × 3^2y = 5184. Step 2: Find the prime factorization of 5184. 5184 = 2 × 2592 2592 = 2 × 1296 1296 = 2 × 648 648 = 2 × 324 324 = 2 × 162 162 = 2 × 81 So, 5184 = 2^6 × 81. Now, factorize 81: 81 = 3 × 27 27 = 3 × 9 9 = 3 × 3 So, 81 = 3^4. Therefore, 5184 = 2^6 × 3^4. Step 3: Equate the powers of the prime factors. Comparing 2^x × 3^2y = 2^6 × 3^4: For the base 2: x = 6 For the base 3: 2y = 4 y = (4)/(2) y = 2 The values are x=6, y=2. Question 13: To evaluate (1.2)/((0.0415)^2) + 5sqrt(0.976) using tables, we would typically look up the square of 0.0415 and the square root of 0.976. As an AI, I will calculate these values directly and then round the final answer to three decimal places. Step 1: Calculate (0.0415)^2. (0.0415)^2 = 0.00172225 Step 2: Calculate (1.2)/((0.0415)^2). (1.2)/(0.00172225) ≈ 696.75709 Step 3: Calculate sqrt(0.976). sqrt(0.976) ≈ 0.9879271 Step 4: Calculate 5sqrt(0.976). 5 × 0.9879271 ≈ 4.9396355 Step 5: Add the two parts and round to three decimal places. 696.75709 + 4.9396355 = 701.6967255 Rounding to three decimal places: 701.697 Question 14: Step 1: Write down the formula for the area of a triangle given two sides and the included angle. The area of a triangle is given by: Area = (1)/(2)ab C where a and b are the lengths of two sides, and C is the angle included between those sides. Step 2: Substitute the given values into the formula. Given: PQ = 5 cm QR = x^2 cm PQR = 30^ Area = 20 cm^2 20 = (1)/(2) × PQ × QR × ( PQR) 20 = (1)/(2) × 5 × x^2 × (30^) Step 3: Evaluate (30^) and simplify the equation. We know that (30^) = (1)/(2). 20 = (1)/(2) × 5 × x^2 × (1)/(2) 20 = (5)/(4) x^2 Step 4: Solve for x^2. Multiply both sides by 4: 20 × 4 = 5x^2 80 = 5x^2 Divide both sides by 5: x^2 = (80)/(5) x^2 = 16 Step 5: Solve for x. Take the square root of both sides. Since x represents a length (part of QR = x^2), it must be positive. x = sqrt(16) x = 4 The value of x is 4. What's next?