A loading dock ramp that is 14 feet long rises at an angle of 14.1° from the horizontal. Due to new design specifications, a longer ramp is to be used, so that the angle is reduced to 9.2°. How much farther out from the dock will that put the foot of the ramp?
the distance out from the dock for an angle θ is just d = 14 cosθ
So, the change in distance is just
14(cos 9.2° - cos 14.1°)
A loading dock ramp that is 14 feet long rises at an angle of 14.1° from the horizontal. Due to new design specifications, a longer ramp is to be used, so that the angle is reduced to 9.2°. How much farther out from the dock will that put the foot of the ramp?
To find out how much farther out from the dock the foot of the ramp will be, we need to calculate the horizontal distance the ramp will move.
Let's start by using the given information for the current ramp angle:
Ramp length = 14 feet
Ramp angle = 14.1°
To calculate the current distance from the dock to the foot of the ramp:
Distance_from_dock = Ramp_length * cos(Ramp_angle)
Distance_from_dock = 14 * cos(14.1°)
Using a calculator, we find:
Distance_from_dock = 14 * 0.9702
Distance_from_dock ≈ 13.58 feet
Now, let's find out the new distance from the dock to the foot of the ramp, with the reduced angle:
New_ramp_angle = 9.2°
To calculate the new distance from the dock to the foot of the ramp, we can use the equation:
New_distance_from_dock = Ramp_length * cos(New_ramp_angle)
New_distance_from_dock = 14 * cos(9.2°)
Using a calculator, we find:
New_distance_from_dock = 14 * 0.9733
New_distance_from_dock ≈ 13.62 feet
The foot of the ramp will be approximately 13.62 feet farther out from the dock with the new ramp angle of 9.2°.
To find out how much farther out from the dock the foot of the ramp will be, we need to first determine the original height of the ramp and then calculate the new length using the given angle. Let's break it down step by step:
Step 1: Find the original height of the ramp
To find the original height of the ramp, we can use the length of the ramp and the angle it makes with the horizontal. Since we know the length of the ramp is 14 feet and the angle is 14.1°, we can use trigonometry.
Using the formula: Height = Length * sin(Angle)
Height = 14 * sin(14.1°)
Height ≈ 3.16 feet (rounded to two decimal places)
Step 2: Calculate the new length of the ramp
Now that we have the original height, we need to find the new length of the ramp using the desired angle of 9.2°. We can use the same formula as before, replacing the angle with the new value.
New Length = Height / sin(New Angle)
New Length = 3.16 / sin(9.2°)
New Length ≈ 21.18 feet (rounded to two decimal places)
Step 3: Find the difference in the position of the foot of the ramp
To determine how much farther out from the dock the foot of the ramp will be, we subtract the original length (14 feet) from the new length (21.18 feet).
Difference = New Length - Original Length
Difference = 21.18 - 14
Difference ≈ 7.18 feet (rounded to two decimal places)
Therefore, the foot of the ramp will be approximately 7.18 feet farther out from the dock when the angle is reduced to 9.2°.