Geometry Help?

Question

A wooden board is placed so that it leans against a loading dock to provide a ramp.

The board is supported by a metal beam perpendicular to the ramp and placed on a 1 ft. tall support.

The bottom of the support is 8 feet from the point where the ramp meets the ground.

The slope of the ramp is 2/5 (this means that for every 2 feet it goes up, it goes 5 to the side).

Find the length of the beam to the nearest hundredth of a foot. Note that the 1 ft. support is vertical, but the metal beam is not.
_____

The answer I got was 2.24 feet.

I took the point where the support meats the beam and
the point where the beam meets the ramp and used the distance formula.

By using those points, I can make a right triangle with a base
of 1 and a side of 2. Using the Pythagorean theorem, 1^2 + 2^2 = c^2, the hypotenuse
of the right triangle is sqrt(5) or 2.24 feet. So, the length of the metal beam
is 2.24 ft

This is a good first try. Please work through this one again, showing me ALL your wor

Answer 1

Okay, I got 2.04.
I first got the angle of ramp, which is arctan(2/5) = 21.8 degrees. Then I found angle made by support which is
arctan(1/8) = 7.125 degrees.
SO, then the beam is sqrt(8^2+1^1) * sin(21.8-7.125) =
2.04 feet approxiamtely.

Answer 2

This can't be right. The right triangle must have sides with a ratio of 2 to 5.

Using trig, you can find the angle of the ramp to the ground is, arc tan 2/5=21°48'05".
The ht. of the ramp at the support is, tan 21°48'05"x8=3.20'.

Vertical distance from support to ramp=3.20-1.00-2.20'.

This is the hypotenuse of a similar triangle, and the sides can be solved for.
Length of beam=cos 21°48'05"x2.20=2.04'.

Answer 3

The bottom of the beam is at (8,1)
The equation of the ramp is y = .4x
The equation of the beam is y =-2.5 x+ b
So, 1 = -2.5*8 +b --> b = 21 so y= -2.5x +21
The inetersection of beam and ramp occurs when:
.4x = -2.5x+21 --> x = 21/2.9 = 7.24
When x = 7.24 , y = .4*7.24 = 2.89
Hence the top of the beam meets the ramp at(7.24,2.89)
length of beam = sqrt((2.89-1)^2 + (7.24-8)^2)= 2.04

I do not see how you got your right triangle with legs of 1 and 2.