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 9 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.
Remember a slope of 2/5 means that everytime you go horizontal 5 units, you go vertical 2 units. (So at 10 units, it should be vertical 4 units, and at 15 it should be vertical 6 units, etc.) Also remember that perpendicular lines have an opposite slope...so if a line has a slope of 1/3, then its perpendicular line will have a slope of -3/1.
Find the distances and plot the points, so you can use the distance formula on the points on either side of the metal beam to find its length! Look for any possible way to find the information you need!
Do the problem step by step, showing your work and explaining what you did to find the answer (the drawings here will help, but make your own--you do not need to submit your drawing, you only need to use it for your own reference). On three of the steps, "prove" what you're doing by stating the theorem or definition you're working with. (You must have three separate proofs--using the distance formula three times doesn't count!)
I am so confused. Please, help.
To find the length of the metal beam, we need to break down the problem into smaller steps.
Step 1: Determine the coordinates of the points where the metal beam intersects the ramp and the support.
Let's assume the point where the ramp meets the ground is the origin (0, 0). Since the slope of the ramp is 2/5, for every 5 units horizontal, we go up by 2 units. Therefore, the coordinates where the ramp intersects the metal beam would be (x, y) = (5n, 2n), where n is a positive integer representing the number of times the slope is applied.
The support is 1 ft. tall, so it intersects the ramp at (0, 1). Since the support is perpendicular to the ramp, the metal beam intersects the support at a right angle. Therefore, the coordinates where the metal beam intersects the support would be (0, 1 + n).
Step 2: Find the value of n.
To find the value of n, we can use the fact that the bottom of the support is 9 feet from the point where the ramp meets the ground. Since the ramp goes up by 2 units for every 5 units horizontal, we know that the x-coordinate of the point where the ramp intersects the metal beam is 9/2 * 5 = 22.5. Therefore, n = 22.5 / 5 = 4.5.
Step 3: Find the coordinates of the points where the metal beam intersects the ramp and the support.
Using the value of n from the previous step, we can find the coordinates of the points of intersection:
- The point where the metal beam intersects the ramp: (x, y) = (5 * 4.5, 2 * 4.5) = (22.5, 9)
- The point where the metal beam intersects the support: (x, y) = (0, 1 + 4.5) = (0, 5.5)
Step 4: Calculate the length of the metal beam using the distance formula.
The length of the metal beam can be calculated using the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the points of intersection:
- Length of the metal beam = √((22.5 - 0)^2 + (9 - 5.5)^2)
= √(506.25 + 13.0625)
= √519.3125
≈ 22.79 feet (rounded to the nearest hundredth)
Therefore, the length of the metal beam is approximately 22.79 feet.