A man drove a car from his office \( (O) \), 120 km on a bearing of \( 135^{\circ} \), and then drove for another 100 km on a bearing of \( 048^{\circ} \) until he reached a town \( (T) \). (a) What is the distance between the (b) Wan's office and the town? (b) is the bearing of \( O \) from \( T \) ?

To solve this problem, we will break it down into steps. ### Step 1: Understand the Bearings and Distances 1. The man drives from his office \( O \) to a point \( A \) which is 120 km on a bearing of \( 135^{\circ} \). 2. Then he drives from point \( A \) to town \( T \) which is 100 km on a bearing of \( 048^{\circ} \). ### Step 2: Convert Bearings to Cartesian Coordinates Bearings are measured clockwise from the north direction. We will convert these bearings into Cartesian coordinates (x, y) using trigonometric functions. 1. **For the first leg (OA)**: - Distance \( d_1 = 120 \) km - Bearing \( \theta_1 = 135^{\circ} \) - Coordinates of point \( A \): \[ A_x = d_1 \cdot \sin(\theta_1) = 120 \cdot \sin(135^{\circ}) \] \[ A_y = d_1 \cdot \cos(\theta_1) = 120 \cdot \cos(135^{\circ}) \] 2. **For the second leg (AT)**: - Distance \( d_2 = 100 \) km - Bearing \( \theta_2 = 048^{\circ} \) - Coordinates of point \( T \): \[ T_x = A_x + d_2 \cdot \sin(\theta_2) = A_x + 100 \cdot \sin(048^{\circ}) \] \[ T_y = A_y + d_2 \cdot \cos(\theta_2) = A_y + 100 \cdot \cos(048^{\circ}) \] ### Step 3: Calculate the Coordinates We will calculate the coordinates of points \( A \) and \( T \) using the above formulas. ### Step 4: Calculate the Distance \( OT \) The distance \( OT \) can be calculated using the distance formula: \[ OT = \sqrt{(T_x - O_x)^2 + (T_y - O_y)^2} \] Since \( O \) is at the origin \( (0, 0) \), this simplifies to: \[ OT = \sqrt{T_x^2 + T_y^2} \] ### Step 5: Calculate the Bearing of \( O \) from \( T \) The bearing of \( O \) from \( T \) can be calculated using the arctangent function: \[ \text{Bearing} = \tan^{-1}\left(\frac{T_x}{T_y}\right) \] We will adjust this angle to get the correct bearing. Now, let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(100\cos\left(48\right)\) Calculate or simplify the expression \( 120 * \sin(135) \). Calculate the value by following steps: - step0: Calculate: \(120\sin\left(135\right)\) Calculate or simplify the expression \( 120 * \cos(135) \). Calculate the value by following steps: - step0: Calculate: \(120\cos\left(135\right)\) Calculate or simplify the expression \( 100 * \sin(48) \). Calculate the value by following steps: - step0: Calculate: \(100\sin\left(48\right)\) Calculate or simplify the expression \( -119.53054 + (-64.014434) \). Calculate the value by following steps: - step0: Calculate: \(-119.53054+\left(-64.014434\right)\) - step1: Remove the parentheses: \(-119.53054-64.014434\) - step2: Subtract the numbers: \(-183.544974\) Calculate or simplify the expression \( 10.604242 + (-76.825466) \). Calculate the value by following steps: - step0: Calculate: \(10.604242+\left(-76.825466\right)\) - step1: Remove the parentheses: \(10.604242-76.825466\) - step2: Subtract the numbers: \(-66.221224\) Calculate or simplify the expression \( \sqrt((-66.221224)^2 + (-183.544974)^2) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\left(-66.221224\right)^{2}+\left(-183.544974\right)^{2}}\) - step1: Convert the expressions: \(\sqrt{\left(-\frac{8277653}{125000}\right)^{2}+\left(-183.544974\right)^{2}}\) - step2: Convert the expressions: \(\sqrt{\left(-\frac{8277653}{125000}\right)^{2}+\left(-\frac{91772487}{500000}\right)^{2}}\) - step3: Add the numbers: \(\sqrt{\frac{16\times 8277653^{2}+91772487^{2}}{500000^{2}}}\) - step4: Use the properties of radicals: \(\frac{\sqrt{16\times 8277653^{2}+91772487^{2}}}{\sqrt{500000^{2}}}\) - step5: Simplify the expression: \(\frac{\sqrt{16\times 8277653^{2}+91772487^{2}}}{500000}\) Calculate or simplify the expression \( \sqrt((-66.221)^2 + (-183.545)^2) \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\left(-66.221\right)^{2}+\left(-183.545\right)^{2}}\) - step1: Convert the expressions: \(\sqrt{\left(-\frac{66221}{1000}\right)^{2}+\left(-183.545\right)^{2}}\) - step2: Convert the expressions: \(\sqrt{\left(-\frac{66221}{1000}\right)^{2}+\left(-\frac{36709}{200}\right)^{2}}\) - step3: Add the numbers: \(\sqrt{\frac{66221^{2}+183545^{2}}{1000^{2}}}\) - step4: Use the properties of radicals: \(\frac{\sqrt{66221^{2}+183545^{2}}}{\sqrt{1000^{2}}}\) - step5: Simplify the expression: \(\frac{\sqrt{66221^{2}+183545^{2}}}{10^{3}}\) - step6: Evaluate the power: \(\frac{\sqrt{66221^{2}+183545^{2}}}{1000}\) It seems that there are persistent issues with calculating the angle using the arctangent function. However, we can still derive the necessary information based on the coordinates. ### Final Results 1. **Distance between the man's office \( O \) and the town \( T \)**: - The distance \( OT \approx 195.13 \) km. 2. **Bearing of \( O \) from \( T \)**: - Since both coordinates \( T_x \) and \( T_y \) are negative, the angle will be in the third quadrant. The bearing can be calculated as: \[ \text{Bearing} = 180 + \tan^{-1}\left(\frac{T_x}{T_y}\right) \] However, since we encountered issues with the calculation, we can estimate that the bearing will be approximately \( 180 + \theta \) where \( \theta \) is the angle calculated from the coordinates. ### Conclusion - **(a)** The distance between the man's office and the town is approximately \( 195.13 \) km. - **(b)** The bearing of \( O \) from \( T \) can be estimated based on the coordinates, but further calculation is needed to provide an exact value. If you would like to proceed with a specific calculation for the bearing or any other aspect, please let me know!