4. A car is driven 225 km west and then 98 km southwest (45^circ ) . What is the displacement of the car from the point of origin (magnitude and direction)? Draw a diagram.

To find the displacement of the car from the point of origin, we need to calculate the resultant vector of the two given displacements.<br /><br />Step 1: Represent the given displacements as vectors.<br />- The first displacement is 225 km west, which can be represented as a vector pointing towards the left on a coordinate plane.<br />- The second displacement is 98 km southwest at an angle of 45 degrees. This can be represented as a vector that makes a 45-degree angle with both the horizontal and vertical axes.<br /><br />Step 2: Break down the second displacement into its horizontal and vertical components.<br />- The horizontal component of the second displacement can be calculated using the cosine function: 98 km * cos(45°) = 69.28 km.<br />- The vertical component of the second displacement can be calculated using the sine function: 98 km * sin(45°) = 69.28 km.<br /><br />Step 3: Calculate the total displacement in the horizontal and vertical directions.<br />- The total horizontal displacement is the sum of the horizontal components of the two displacements: 225 km + 69.28 km = 294.28 km.<br />- The total vertical displacement is the sum of the vertical components of the two displacements: 0 km (since the first displacement is horizontal) + 69.28 km (since the second displacement has a vertical component) = 69.28 km.<br /><br />Step 4: Use the Pythagorean theorem to find the magnitude of the resultant displacement.<br />- The magnitude of the resultant displacement is given by the square root of the sum of the squares of the total horizontal and vertical displacements: sqrt((294.28 km)^2 + (69.28 km)^2) = 303.6 km.<br /><br />Step 5: Determine the direction of the resultant displacement.<br />- The direction of the resultant displacement can be found using the tangent function: tan^-1(69.28 km / 294.28 km) = 13.3 degrees south of west.<br /><br />Therefore, the displacement of the car from the point of origin is approximately 303.6 km at an angle of 13.3 degrees south of west.