I have this much so far
(1+x)/9 = 2/5 or x = 13/5 = 2.6 ft
I am unsure of how to work out the rest of the problem.
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.
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!
Cheers!
To solve this problem, we need to determine the length of the metal beam supporting the wooden board. We have the information about the height of the support and the distance from the support to the point where the ramp meets the ground.
Let's break down the problem step-by-step:
Step 1: Draw a diagram
First, let's draw a diagram to visualize the problem. Sketch a right-angled triangle, representing the ramp leaning against the loading dock, with the support (1 ft. tall) at the base of the triangle. Label the distance from the support to the ground as 9 feet and mark the slope of the ramp (2/5) on the diagram.
Step 2: Find the length of the ramp
From the given slope (2/5), we know that for every 2 feet up, the ramp goes 5 feet to the side. This means that the vertical height of the ramp is 2 feet for every 5 feet of horizontal distance. We can scale this proportion to find the length of the ramp.
Let x be the horizontal distance from the support to the point where the ramp meets the loading dock. Since we know the slope is 2/5, the vertical height of the ramp is (2/5)x.
Given that the total height from the support to the loading dock is 1 ft, we can set up the following equation:
(2/5)x + 1 = 9
Step 3: Solve the equation
To find x, the distance from the support to the point where the ramp meets the loading dock, we can solve the equation.
Start by subtracting 1 from both sides of the equation:
(2/5)x = 9 - 1
Simplifying:
(2/5)x = 8
To isolate x, multiply both sides by 5/2:
x = (5/2) * 8
x = 20
Therefore, the distance from the support to the point where the ramp meets the loading dock (x) is 20 feet.
Step 4: Find the length of the metal beam
Now that we have the distance from the support to the point where the ramp meets the loading dock, we can use the distance formula to find the length of the metal beam.
The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In our case, the two points are the support (0, 0) and the point where the ramp meets the loading dock (20, 1). Substituting these values into the distance formula, we get:
d = sqrt((20 - 0)^2 + (1 - 0)^2)
d = sqrt(20^2 + 1^2)
d = sqrt(400 + 1)
d = sqrt(401)
Using a calculator or a math tool, we find that the length of the metal beam is approximately 20.02 feet to the nearest hundredth.
Therefore, the length of the beam is approximately 20.02 feet, rounded to the nearest hundredth.