Determining True Bearing

The initial magnetic bearing (MB) in 1990 was N 40°30′ W. The magnetic declination in 1990 was 3°30′ E. Magnetic declination indicates the angle between True North and Magnetic North. An Easterly declination means Magnetic North is East of True North.

To find the True Bearing (TB), we adjust the Magnetic Bearing using the declination. Since the bearing is measured West from Magnetic North (N 40°30′ W) and Magnetic North is East of True North (3°30′ E), the angle measured West from True North will be the difference:

$ TB_{\text{angle}} = MB_{\text{angle}} - \text{Declination}_{\text{East}} $

$ TB_{\text{angle}} = 40^\circ 30' - 3^\circ 30' = 37^\circ 00' $

The True Bearing remains measured from North towards West, so it is N 37°00′ W.

Calculating New Magnetic Bearing

The True Bearing (N 37°00′ W) is constant. The present magnetic declination is 2°10′ W. A Westerly declination means Magnetic North is West of True North.

We need to find the new magnetic bearing (MB_new) relative to the current Magnetic North. The line is 37°00′ West of True North. The current Magnetic North is 2°10′ West of True North.

The angle measured from the current Magnetic North towards West will be:

$ MB_{\text{new angle}} = TB_{\text{angle}} - \text{Declination}_{\text{West}} $

$ MB_{\text{new angle}} = 37^\circ 00' - 2^\circ 10' = 34^\circ 50' $

Since the True Bearing was West of North, the new Magnetic Bearing is also West of North.

The new magnetic bearing is N 34°50′ W.

Final Result

The calculated new magnetic bearing is N 34°50′ W, which corresponds to the Reduced Bearing system. This matches Option 4.