PLEASE! Someone help me with geometry!?
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!)
Cheers!
To solve this problem, we can break it down into smaller steps. Let's start by drawing a diagram to represent the situation described.
1. Draw a straight line to represent the ground.
2. Mark a point on the line representing the point where the ramp meets the ground.
3. Measure 1 foot vertically upward from this point and draw a vertical line to represent the support.
4. Mark a point on the support line that is 9 feet horizontally away from the point where the ramp meets the ground.
5. Connect the top of the support line to the point where the ramp meets the ground to represent the wooden board or ramp.
6. Remember that the slope of the ramp is 2/5, which means that for every 2 feet it goes up, it goes 5 feet to the side. Use this information to draw the ramp appropriately.
Now, we need to find the length of the metal beam. Let's proceed step by step.
Step 1: Identify the points on either side of the metal beam.
Look at the diagram and mark two points - one at the top of the support line and the other at the bottom of the ramp where it meets the ground. These two points represent the endpoints of the metal beam.
Step 2: Find the coordinates of the identified points.
Using the information from the diagram, let's find the coordinates of the two points.
- The point at the top of the support line has coordinates (9, 1) since it is 9 feet horizontally away from the point where the ramp meets the ground and 1 foot vertically above it.
- The point at the bottom of the ramp has coordinates (0, 0) since it is the origin of our coordinate system.
Step 3: Calculate the distance between the two points using the distance formula.
The distance formula helps us find the length of a straight line between two points in a coordinate plane. The formula is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Substituting the coordinates of the two points into the formula:
Distance = √((0 - 9)^2 + (0 - 1)^2)
= √((-9)^2 + (-1)^2)
= √(81 + 1)
= √82
Therefore, the length of the metal beam to the nearest hundredth of a foot is approximately 9.06 feet.
Proof:
The distance formula used in Step 3 is derived from the Pythagorean Theorem, a fundamental theorem in geometry. The formula states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In our case, the two sides of the right triangle are the horizontal distance (9 units) and the vertical distance (1 unit). Calculating the square of both distances and then adding them gives us 82. Finally, taking the square root of 82 gives us the length of the metal beam (9.06 feet). This demonstrates the application of the Pythagorean Theorem and the distance formula.