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 are right to point out the curve equation, Desmond. Let's clarify the situation. The problem states that a point is moving along the intersection of two equations: 1. Curve: x^2 + 3xy + 3y^2 = 22 2. Line: x - 2y + 4 = 0 We are asked to find the speed of this point when x=2, given (dx)/(dt) = 3 units/second. First, let's check if there is a point on the intersection where x=2. If x=2, from the line equation x - 2y + 4 = 0: 2 - 2y + 4 = 0 6 - 2y = 0 2y = 6 y = 3 So, the point (2, 3) lies on the line. Now, let's check if this point (2, 3) also lies on the curve x^2 + 3xy + 3y^2 = 22: (2)^2 + 3(2)(3) + 3(3)^2 = 4 + 18 + 27 = 49 Since 49 ≠ 22, the point (2, 3) is not on the curve. This means there is no point that satisfies both the curve equation and the line equation at x=2. Therefore, a point cannot be "moving along the intersection" when x=2. The problem statement contains an inconsistency. To provide a solution, we must make a reasonable assumption. The most common approach in such cases is to assume the problem intends for the point to be moving along one of the given paths, and the other information is either extraneous or there's a typo in the problem. Given that the line equation directly relates x and y in a simpler way, we will assume the point is moving along the line x - 2y + 4 = 0 at the instant x=2. Step 1: Find the y-coordinate when x=2 using the line equation. Given the line equation x - 2y + 4 = 0. Substitute x=2: 2 - 2y + 4 = 0 6 - 2y = 0 2y = 6 y = 3 So, at the instant x=2, the point is (2, 3). Step 2: Differentiate the line equation with respect to time t. The equation of the line is x - 2y + 4 = 0. Differentiate both sides with respect to t: (d)/(dt)(x - 2y + 4) = (d)/(dt)(0) (dx)/(dt) - 2(dy)/(dt) + 0 = 0 (dx)/(dt) - 2(dy)/(dt) = 0 Step 3: Substitute the given rate (dx)/(dt) to find (dy)/(dt). We are given (dx)/(dt) = 3 units/second. 3 - 2(dy)/(dt) = 0 2(dy)/(dt) = 3 (dy)/(dt) = (3)/(2) units/second Step 4: Calculate the speed of the point. The speed of a point (x(t), y(t)) in 2D is given by the formula: Speed = sqrt(((dx)/(dt))^2 + ((dy)/(dt))^2) Substitute the values of (dx)/(dt) = 3 and (dy)/(dt) = (3)/(2): Speed = sqrt((3)^2 + ((3)/(2))^2) Speed = sqrt(9 + (9)/(4)) Speed = sqrt((36)/(4) + (9)/(4)) Speed = sqrt((45)/(4)) Speed = sqrt(45)sqrt(4) Speed = sqrt(9 × 5)2 Speed = 3sqrt(5)2 units/second The speed of the point is 3sqrt(5)2 units/second. 3 done, 2 left today. You're making progress.