A ladder leans against a vertical wall and makes an angle of 76˚ with the ground. The foot of the ladder is 1.8 m from the base of the wall. Determine the vertical distance from the ground to the top of the ladder, to the nearest tenth of a metre
To determine the vertical distance from the ground to the top of the ladder, we can use trigonometry.
Let x be the vertical distance from the ground to the top of the ladder.
Since the ladder leans against a vertical wall and makes an angle of 76˚ with the ground, we can use the sine function to set up the following equation:
sin(76˚) = x / hypotenuse,
where the hypotenuse is the length of the ladder.
We can find the length of the ladder using the Pythagorean Theorem.
Let h be the length of the ladder. Then:
h^2 = (1.8)^2 + x^2,
h^2 = 3.24 + x^2.
Now, we substitute this expression for h^2 into the trigonometric equation:
sin(76˚) = x / √(3.24 + x^2).
Solving for x, we get:
x = √(3.24 + x^2) * sin(76˚),
x = √(3.24 + x^2) * 0.9781,
x = 0.9781√(3.24 + x^2).
Squaring both sides and rearranging:
x^2 = 0.9566^2 * (3.24 + x^2),
x^2 = 0.9566^2 * 3.24 + 0.9566^2 * x^2,
x^2 = 2.9966 + 0.9566^2 * x^2,
0.0434x^2 = 2.9966,
x^2 = 2.9966 / 0.0434,
x^2 = 69.0194,
x ≈ √69.0194,
x ≈ 8.3 m.
Therefore, the vertical distance from the ground to the top of the ladder is approximately 8.3 metres.