A ladder leaning against a wall makes an angle of 60 degree with the horizontal If the foot of the ladder is 2.5 m away from the wall , find the length of the ladder.

recall the sides ratios for a 30-60-90 right triangle:

1:√3:2

5m

To find the length of the ladder, we can use trigonometry. In this case, we can use the sine function since we have the opposite side length and the hypotenuse. Here are the steps to solve the problem:

1. Draw a right triangle with the ladder as the hypotenuse, the wall as the adjacent side, and the ground as the opposite side.

2. Label the sides of the triangle - the hypotenuse as "x" (which represents the length of the ladder), the adjacent side as 2.5 m, and the angle opposite to the adjacent side as 60 degrees.

3. Use the sine function to relate the opposite and hypotenuse sides:
sin(60°) = opposite/hypotenuse
sin(60°) = 2.5/x

4. Solve for x by cross-multiplying:
(2.5/x) = sin(60°)
2.5 = x * sin(60°)

5. Divide both sides by sin(60°) to isolate x:
x = 2.5 / sin(60°)

6. Calculate the value of sin(60°) using a calculator:
sin(60°) = 0.866

7. Substitute the value of sin(60°) into the equation:
x = 2.5 / 0.866

8. Calculate the value of x using a calculator:
x ≈ 2.89

Therefore, the length of the ladder is approximately 2.89 meters.