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: Write the variation equation. The problem states that y varies inversely as the square of x. This can be written as: y = (k)/(x^2) where k is the constant of proportionality. Step 2: Use the given information to find the constant k. We are given that the difference between the values of y when x=6 and x=10 is 16. When x=6, y_1 = (k)/(6^2) = (k)/(36). When x=10, y_2 = (k)/(10^2) = (k)/(100). Since y varies inversely with x^2, a smaller x value will result in a larger y value. Thus, y_1 > y_2. The difference is y_1 - y_2 = 16. (k)/(36) - (k)/(100) = 16 To solve for k, find a common denominator for 36 and 100. The least common multiple of 36 and 100 is 900. (25k)/(900) - (9k)/(900) = 16 (25k - 9k)/(900) = 16 (16k)/(900) = 16 Multiply both sides by 900: 16k = 16 × 900 Divide both sides by 16: k = 900 Step 3: State the law connecting x and y. Substitute the value of k back into the variation equation from Step 1: y = (900)/(x^2) Step 4: Find x when y=4. Substitute y=4 into the law found in Step 3: 4 = (900)/(x^2) Multiply both sides by x^2: 4x^2 = 900 Divide both sides by 4: x^2 = (900)/(4) x^2 = 225 Take the square root of both sides: x = ±sqrt(225) x = ± 15 The law connecting x and y is y = (900)/(x^2). When y=4, x = ± 15. Send me the next one 📸