The bottom of the doorway to a building is 1.55 ft above the ground, and a ramp to the door for the disabled is at an angle of 4.0 degrees with the ground. How much longer must the ramp be in order to make the angle 2.0 degrees?

To find out how much longer the ramp must be in order to decrease the slope from 4.0 degrees to 2.0 degrees while maintaining the height at 1.55 ft, we will use trigonometry. The height of the ramp remains constant at 1.55 ft, which acts as the opposite side of a right triangle. We have two angles (4.0 degrees and 2.0 degrees) corresponding to two different lengths of the hypotenuse (the ramp).

For the first scenario with a 4.0 degree angle, we use the tangent function:

Tan(4.0 degrees) = Opposite / Adjacent, which is the ramp's height over its base length.
Rearranging, Adjacent = Opposite / Tan(4.0 degrees).
Thus, Base length at 4.0 degrees = 1.55 ft / Tan(4.0 degrees).

For the second scenario with a 2.0 degree angle:

Tan(2.0 degrees) = Opposite / Adjacent.
Rearranging, Adjacent = Opposite / Tan(2.0 degrees).
Thus, Base length at 2.0 degrees = 1.55 ft / Tan(2.0 degrees).

Using a calculator, we find:

Base length at 4.0 degrees ≈ 22.21 ft (using Tan(4.0 degrees) ≈ 0.06993).
Base length at 2.0 degrees ≈ 44.48 ft (using Tan(2.0 degrees) ≈ 0.03492).

To find the additional length needed for the ramp, subtract the base length at 4.0 degrees from the base length at 2.0 degrees. Additional length = 44.48 ft - 22.21 ft ≈ 22.27 ft. The ramp must be approximately 22.27 ft longer to achieve a 2.0 degree angle.