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?
tan 4.5° = .0787
so, you want 1/d <= .0787, or
d >= 12.7
that is, the ramp must extend at least 12.7 m
Well, it seems like we've got some math to do! Now, let's see if I can ramp up the humor while answering this.
So, to find the minimum distance from the entrance that the ramp must begin, we need to consider the angle limitation. We know the maximum angle can be 4.5°, and we want to solve for the minimum distance. Let's get rolling!
To calculate the minimum distance, we can use a bit of trigonometry. By forming a right triangle with the angle of 4.5° as one of the acute angles, we can use the tangent function.
Now, the tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the opposite side is the height difference of 1 m, and the adjacent side is what we're looking for.
Plugging those values into our trusty calculator, we can find that the minimum distance for the ramp to begin is approximately 13.3 meters. So, there you have it – a ramp-tastic solution to the problem!
Remember, safety first, so let's make sure those wheelchair users can make a smooth entrance!
To determine the minimum distance from the entrance that the ramp must begin, you need to calculate the length of the ramp using the given angle of inclination. Here are the steps to find the minimum distance:
Step 1: Convert the angle of inclination from degrees to radians.
To convert degrees to radians, use the formula: radians = (π/180) * degrees.
In this case, the angle is 4.5°, so the conversion would be:
radians = (π/180) * 4.5 = 0.07854 radians (rounded to 5 decimal places).
Step 2: Determine the height of the ramp using trigonometry.
The trigonometric function for calculating the height of the ramp is:
height = total rise * sin(angle of inclination).
In this case, the total rise is 1 m. Using the converted angle from Step 1, the calculation becomes:
height = 1 * sin(0.07854) ≈ 0.07837 m (rounded to 5 decimal places).
Step 3: Calculate the length of the ramp using the height and the angle of inclination.
The length of the ramp can be found using the formula:
length = height / sin(angle of inclination).
Plugging in the values we have, the calculation becomes:
length = 0.07837 / sin(0.07854) ≈ 1.0003 m (rounded to 4 decimal places).
Step 4: Subtract the length of the ramp from the total rise to find the minimum distance.
The minimum distance from the entrance that the ramp must begin is the difference between the total rise and the length of the ramp.
minimum distance = total rise - length = 1 - 1.0003 = -0.0003 m.
Please note that the result of the minimum distance is approximately -0.0003 m, indicating that the ramp would begin -0.0003 meters (or 0.3 mm) below the entrance. This means that the ramp should ideally start at the same level as the front entrance or slightly above it to ensure accessibility and compliance with the city ordinance.
To find the minimum distance from the entrance that the ramp must begin, we need to use trigonometry. The angle of inclination (θ) is given as 4.5°.
Let's assume the minimum distance from the entrance to the start of the ramp is 'x'. The height of the rise is given as 1 m.
We can use the trigonometric relationship between the angle of inclination, the height, and the horizontal distance to find the value of 'x'. In this case, we will use the tangent function:
tan(θ) = opposite/adjacent
Here, the opposite side is the height (1 m), and the adjacent side is 'x'. We can rearrange the equation to solve for 'x':
x = opposite/tan(θ)
Plugging in the values, we get:
x = 1/tan(4.5°)
Now, we can calculate this value:
x ≈ 12.66 meters
Therefore, the minimum distance from the entrance that the ramp must begin is approximately 12.66 meters.