a ladder leaning against a wall makes a 60 degree angle with the ground. the based of the ladder is 4m from the building. how high above the ground is the top of the ladder
hint: tan60° = √3
h/4 = tan60°
To find the height above the ground of the top of the ladder, we can use trigonometry, specifically the sine function.
Let's denote the height above the ground as 'h'. We know that the base of the ladder forms a right triangle with the ground and the wall. The angle between the ladder and the ground is given as 60 degrees.
We can use the sine function to find the height:
sin(angle) = opposite / hypotenuse
In this case, the opposite side is the height 'h' and the hypotenuse is the length of the ladder. The length of the ladder can be calculated using the Pythagorean theorem:
ladder^2 = base^2 + height^2
Substituting the given values, we have:
ladder^2 = 4^2 + h^2
Now, let's solve for the ladder length:
ladder^2 = 16 + h^2
To find the length of the ladder, we take the square root of both sides:
ladder = √(16 + h^2)
Since we know the ladder forms a 60-degree angle with the ground, we can calculate the sine of the angle as sin(60°) = √3/2:
√3/2 = h / ladder
We can rearrange this equation to solve for 'h':
h = (√3/2) * ladder
Now, substitute the value of ladder from the previous equation:
h = (√3/2) * √(16 + h^2)
Simplifying further:
h = (√3/2) * √16 * √(1 + h^2/16)
h = √3 * √(1 + h^2/16)
Squaring both sides:
h^2 = 3 * (1 + h^2/16)
h^2 = 3 + 3h^2/16
16h^2 = 48 + 3h^2
13h^2 = 48
h^2 = 48/13
Taking the square root of both sides:
h ≈ √(48/13)
h ≈ 3.68 meters (rounded to two decimal places)
Therefore, the top of the ladder is approximately 3.68 meters above the ground.