If u is a function of v such that u is a constant plus a term that varies directly as the square of v. If u = 35 when v = 2, and u = 203 when v = 5, find v when u = 515.

Step 1: Formulate the equation for u. The problem states that u is the sum of two terms: one is a constant, and the other varies directly as the square of v. We can write this relationship as: u = A + Bv^2 where A is the constant term and B is the constant of proportionality. Step 2: Use the first set of given values to form an equation. When v = 2, u = 35. Substitute these values into the equation: 35 = A + B(2)^2 35 = A + 4B (*) Step 3: Use the second set of given values to form another equation. When v = 5, u = 203. Substitute these values into the equation: 203 = A + B(5)^2 203 = A + 25B (**) Step 4: Solve the system of linear equations to find A and B. Subtract equation () from equation (*): (A + 25B) - (A + 4B) = 203 - 35 21B = 168 Divide by 21: B = (168)/(21) B = 8 Substitute B = 8 into equation (*): 35 = A + 4(8) 35 = A + 32 A = 35 - 32 A = 3 So, the relationship between u and v is u = 3 + 8v^2. Step 5: Find the value of v when u = 515. Substitute u = 515 into the derived equation: 515 = 3 + 8v^2 Subtract 3 from both sides: 515 - 3 = 8v^2 512 = 8v^2 Divide by 8: v^2 = (512)/(8) v^2 = 64 Take the square root of both sides. Since v typically represents a positive quantity in such problems, we take the positive root: v = sqrt(64) v = 8 The value of v when u = 515 is 8. Send me the next one 📸