A ladder leaning against a wall makes an angle 60 with the horizontal if the foot of the ladder is the 2.5m away from the wall find the length of the ladder
recall the sides ratios for a 30-60-90 right angle
1:√3:2
To find the length of the ladder, we can use some trigonometry.
Let's label the length of the ladder as "L". We know that the foot of the ladder is 2.5m away from the wall, and the angle between the ladder and the horizontal is 60 degrees.
The ladder, the wall, and the ground form a right-angled triangle. The ladder is the hypotenuse of this triangle, the distance from the foot of the ladder to the wall is the base, and the height is the distance from the top of the ladder to the ground.
We can use the trigonometric function "cosine" to calculate the length of the ladder. Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
In this case, the adjacent side is the distance from the foot of the ladder to the wall (2.5m), and the hypotenuse is the length of the ladder (L). So we have:
Cos(60°) = Adjacent / Hypotenuse
Cos(60°) = 2.5 / L
Now, we can solve for L by rearranging the equation:
L = 2.5 / Cos(60°)
Using the formula for the cosine of 60 degrees, which is √3/2, we can further simplify the equation:
L = 2.5 / (√3/2)
To simplify this expression, we multiply the numerator and denominator by 2:
L = (2.5 * 2) / √3
L = 5 / √3
To rationalize the denominator, we multiply the numerator and denominator by √3:
L = (5 * √3) / (√3 * √3)
L = 5√3 / 3
Therefore, the length of the ladder is approximately 5√3 / 3 meters.