Application The steps to the front entrance of a public building rise a total of 1 m. A portion of the steps will be replaced by a wheelchair ramp. By a city ordinance, the angle of inclination for a ramp cannot measure greater than 4.5°. What is the minimum distance from the entrance that the ramp must begin?

To find the minimum distance from the entrance that the ramp must begin, we need to use trigonometry involving the angle of inclination and the height of the steps.

Let's break down the problem into steps:

Step 1: Calculate the height of the steps in meters.
Given that the total height of the steps is 1 meter.

Step 2: Convert the angle of inclination to radians.
The angle of inclination is given as 4.5 degrees. To convert it to radians, use the formula:
Angle in radians = (Angle in degrees * π) / 180.
So, the angle in radians would be (4.5 * π) / 180.

Step 3: Use trigonometry to find the minimum distance.
We can use the tangent function to relate the angle of inclination and the height of the steps.
Tangent of the angle of inclination = height of the steps / minimum distance of the ramp.
So, the formula is:
tan(angle of inclination) = height of the steps / minimum distance of the ramp.

Step 4: Solve for the minimum distance of the ramp.
Rearranging the formula from Step 3, we get:
minimum distance of the ramp = height of the steps / tan(angle of inclination).

Now, let's calculate the minimum distance of the ramp:

Step 1: Given that the height of the steps is 1 meter.

Step 2: Converting the angle of inclination to radians:
angle in radians = (4.5 * π) / 180 = 0.07854 radians (approx).

Step 3: Using trigonometry:
tan(0.07854) = 1 / minimum distance of the ramp.

Step 4: Solving for the minimum distance of the ramp:
minimum distance of the ramp = 1 / tan(0.07854).

Calculating the minimum distance of the ramp using a calculator, we get:
minimum distance of the ramp ≈ 12.732 meters (approx).

So, the ramp must begin at a minimum distance of approximately 12.732 meters from the entrance.