To solve this problem, we need to find two things:
The length of each leg of the stool.
The distance of the steps from each other.
1. Length of Each Leg
The stool has a square top with a side of 0.5 meters. The height of the stool is 1.5 meters, and each leg is inclined at an angle of 60 degrees to the ground. We need to determine the length of each leg.
Consider the triangle formed by one of the stool legs. This triangle is a right triangle where:
The vertical height of the stool (1.5 meters) is the opposite side.
The length of the leg is the hypotenuse.
The angle of inclination (60 degrees) is between the leg and the ground.
We can use trigonometry to find the length of the leg. The relationship between the opposite side (height) and the hypotenuse (leg length) is given by the sine function:
sin(θ) = opposite / hypotenuse
Here, θ = 60 degrees, opposite = 1.5 meters, and we need to find the hypotenuse (length of the leg).
sin(60) = 1.5 / length of leg
Since sin(60) = √3/2, we can substitute this value:
√3/2 = 1.5 / length of leg
Now, solving for the length of the leg:
length of leg = 1.5 / (√3/2)
length of leg = 1.5 × (2/√3)
length of leg = 3 / √3
length of leg = √3 meters
So, the length of each leg is √3 meters, which is approximately 1.732 meters.
2. Distance Between the Two Steps
The problem mentions that two steps should be placed at equal distances from each other. The height of the stool is 1.5 meters, and the stool has a square top with side 0.5 meters. The steps are at the same height as the stool legs.
Each step must be placed at the same distance from the vertical center of the stool top, and they must be aligned in a way that the legs remain inclined at 60 degrees. To place the steps, we need to look at the horizontal distance covered by the leg when inclined at 60 degrees.
Using trigonometry, we can calculate the horizontal distance (the adjacent side of the triangle):
cos(60) = adjacent / hypotenuse
cos(60) = 1/2
So, the horizontal distance covered by the leg is:
adjacent = length of leg × cos(60)
adjacent = √3 × 1/2
adjacent = √3 / 2 meters
Thus, the distance from the center to one of the steps is approximately 0.866 meters. Since the steps are placed at equal distances, the total distance between the two steps will be twice this distance:
Distance between the two steps = 2 × 0.866 = 1.732 meters
Final Answer:
The length of each leg of the stool is approximately 1.732 meters.
The distance between the two steps is approximately 1.732 meters.
