Question

A ramp is needed to allow vehicles to climb a 2 foot wall. The angle of elevation in order for the vehicles to safely go up must be 30 o or less, and the longest ramp available is 5 feet long. Can this ramp be used safely

Answer 1

Using trigonometry, it was determined that the minimum ramp length needed to climb a 2-foot wall at a 30-degree angle is 4 feet. Therefore, a 5-foot long ramp is safe and exceeds the minimum length required.

Step-by-step explanation:

To determine if a 5-foot long ramp can be used safely to climb a 2-foot wall, with the angle of elevation not exceeding 30 degrees, one needs to use trigonometry. By taking the angle of elevation to be 30 degrees and knowing the opposite side of the right triangle (the height of the wall, which is 2 feet), we can solve for the hypotenuse - the ramp length required. The trigonometric function to use is the sine, which is equal to the opposite side divided by the hypotenuse.

If we set up the equation: sin(30°) = 2 / Hypotenuse, the hypotenuse (H) can be solved as H = 2 / sin(30°). Since sin(30°) = 1/2, the equation simplifies to H = 2 / (1/2), which gives us H = 4 feet as the minimum ramp length needed. Since the available ramp is 5 feet long, it exceeds the minimum length required, making it safe to use based on length and angle criteria.

Answer 2

Yes the ramp can be safely used

Step-by-step explanation:

Here, we have

Length of longest ramp = 5 ft

Height of wall = 2 ft

Therefore, the sine of the angle adjacent to the ramp which is equal to the angle of elevation is given by;

`Sin\theta = (Opposite \, side \, to\, angle)/(Hypothenus\, side \, of\, triangle)`

Where:

The opposite side to angle = 2 ft wall and

Hypotenuse side = Ramp = 5 ft

Therefore,

`Sin\theta = (2)/(5) = 0.4` and θ = sin⁻¹0.4 = 23.55 °

The ramp can be safely used as the angle it is adjacent to is less than the specified 30°.