Find the slope of the power line.

Here are the solutions to your tasks: a) Determine the equation of the perpendicular path from the workshop to the power line. Step 1: Find the slope of the power line. The equation of the power line is 3x + 4y - 24 = 0. Rearrange to slope-intercept form y = mx + c: 4y = -3x + 24 y = -(3)/(4)x + 6 The slope of the power line, m_1, is -(3)/(4). Step 2: Find the slope of the perpendicular path. For perpendicular lines, the product of their slopes is -1. m_1 · m_2 = -1 -(3)/(4) · m_2 = -1 m_2 = (4)/(3) The slope of the perpendicular path, m_2, is (4)/(3). Step 3: Determine the equation of the perpendicular path. The path passes through the workshop at (2,2) and has a slope of (4)/(3). Use the point-slope form y - y_1 = m(x - x_1): y - 2 = (4)/(3)(x - 2) Multiply by 3 to clear the fraction: 3(y - 2) = 4(x - 2) 3y - 6 = 4x - 8 Rearrange to the general form Ax + By + C = 0: 4x - 3y - 2 = 0 The equation of the perpendicular path is 4x - 3y - 2 = 0. b) Calculate the coordinates of the point on the power line where the connection will meet. Step 1: Solve the system of equations for the power line and the perpendicular path. Equation of power line: 3x + 4y = 24 (1) Equation of perpendicular path: 4x - 3y = 2 (2) Step 2: Use the elimination method. Multiply equation (1) by 3 and equation (2) by 4 to eliminate y: 3 × (3x + 4y = 24) 9x + 12y = 72 4 × (4x - 3y = 2) 16x - 12y = 8 Add the two new equations: (9x + 12y) + (16x - 12y) = 72 + 8 25x = 80 x = (80)/(25) = (16)/(5) Step 3: Substitute the value of x into one of the original equations to find y. Using equation (2): 4((16)/(5)) - 3y = 2 (64)/(5) - 3y = 2 3y = (64)/(5) - 2 3y = (64)/(5) - (10)/(5) 3y = (54)/(5) y = (54)/(15) = (18)/(5) The coordinates of the intersection point are ((16)/(5), (18)/(5)) or (3.2, 3.6). c) Use the perpendicular distance formula to find the shortest distance from the workshop to the line. Step 1: Identify the point and the line equation. The workshop is at (x_0, y_0) = (2,2). The power line equation is 3x + 4y - 24 = 0, so A=3, B=4, C=-24. Step 2: Apply the perpendicular distance formula. The formula is D = (|Ax_0 + By_0 + C|)/(sqrt(A^2 + B^2)). D = (|3(2) + 4(2) - 24|)/(sqrt(3^2 + 4^2)) D = (|6 + 8 - 24|)/(sqrt(9 + 16)) D = (|14 - 24|)/(sqrt(25)) D = (|-10|)/(5) D = (10)/(5) D = 2 The shortest distance from the workshop to the power line is 2 units. d) What is the total cost to reach the power line? Step 1: Use the shortest distance and the cost per unit. Shortest distance = 2 units. Cost per unit = 1000. Step 2: Calculate the total cost. Total cost = Shortest distance × Cost per unit Total cost = 2 × 1000 Total cost = 2000 The total cost to reach the power line is 2000. Send me the next one 📸