There is a 3-meter tall vertical ladder from the ground to a ramp. The bottom of the ladder is 10 m from the base of the ramp. The distance along the ramp from the top of the ladder to the loading platform is 24 m. How high above the ground is the loading platform?
To find the height of the loading platform above the ground, we can use the concept of similar triangles. Let's break down the problem step by step:
Step 1: Draw a diagram to visualize the situation.
- Draw a vertical line to represent the ladder, measuring 3 meters.
- Draw a horizontal line for the ground level.
- Draw a ramp starting from the base of the ladder and extending horizontally.
- Label the distance from the bottom of the ladder to the base of the ramp as 10 m.
- Label the distance from the top of the ladder to the loading platform as 24 m.
Step 2: Identify the similar triangles.
- We have two similar triangles: the larger triangle formed by the ground, ladder, and loading platform, and the smaller triangle formed by the ground, ladder, and the base of the ramp.
Step 3: Set up a proportion.
- We can set up a proportion using the corresponding sides of the similar triangles.
- Let "x" represent the height of the loading platform above the ground.
- In the larger triangle, the corresponding side to the ladder is x meters.
- In the smaller triangle, the corresponding side to the ladder is 10 meters.
- So, we can set up the proportion: x/3 = 10/24.
Step 4: Solve the proportion.
- Cross multiply: 24x = 3 * 10.
- Simplify: 24x = 30.
- Divide both sides by 24: x = 30/24.
- Simplify: x = 1.25.
Therefore, the loading platform is 1.25 meters above the ground.