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
Here's the solution for question 1. 1. (a) Find the distance between the points A(-3, 2) and B(4, -6). Step 1: State the distance formula. The distance D between two points (x_1, y_1) and (x_2, y_2) is given by: D = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) Step 2: Substitute the coordinates of points A and B into the formula. Given A(-3, 2) and B(4, -6), we have x_1 = -3, y_1 = 2, x_2 = 4, y_2 = -6. D = sqrt((4 - (-3))^2 + (-6 - 2)^2) D = sqrt((4 + 3)^2 + (-8)^2) D = sqrt((7)^2 + (-8)^2) D = sqrt(49 + 64) D = sqrt(113) Step 3: Calculate the numerical value. D ≈ 10.6301 The distance is: 10.63 units 1. (b) Find the acute angle between the lines 2x + 3y = 5 and 3x + 2y = 1 correct to one decimal place. Step 1: Find the slopes of both lines. For the first line, 2x + 3y = 5: 3y = -2x + 5 y = -(2)/(3)x + (5)/(3) So, the slope m_1 = -(2)/(3). For the second line, 3x + 2y = 1: 2y = -3x + 1 y = -(3)/(2)x + (1)/(2) So, the slope m_2 = -(3)/(2). Step 2: Use the formula for the angle between two lines. The acute angle between two lines with slopes m_1 and m_2 is given by: = | (m_1 - m_2)/(1 + m_1 m_2) | Substitute m_1 = -(2)/(3) and m_2 = -(3)/(2): = | (-2)/(3) - (-(3)/(2))1 + (-(2)/(3))(-(3)/(2)) | = | (-2)/(3) + (3)/(2)1 + (6)/(6) | = | (-4)/(6) + (9)/(6)1 + 1 | = | (5)/(6)2 | = (5)/(6) × (1)/(2) = (5)/(12) Step 3: Calculate the angle . = ((5)/(12)) ≈ 22.61986^ Round to one decimal place: = 22.6^ 3 done, 2 left today. You're making progress.