7. A bdder of length 6 m bass against a wall. The foot of the bodder is at a distance of 3 m from the base of the wall. Calculate the angle between the ladder and the ground.

To find the angle between the ladder and the ground, we can use trigonometry. We have a right triangle formed by the ladder, the wall, and the ground. 1. **Identify the sides of the triangle:** - The length of the ladder (hypotenuse) = 6 m - The distance from the wall to the foot of the ladder (adjacent side) = 3 m - The height of the ladder against the wall (opposite side) can be found using the Pythagorean theorem. 2. **Use the Pythagorean theorem to find the height (opposite side):** \[ \text{height}^2 + \text{base}^2 = \text{hypotenuse}^2 \] \[ \text{height}^2 + 3^2 = 6^2 \] \[ \text{height}^2 + 9 = 36 \] \[ \text{height}^2 = 36 - 9 = 27 \] \[ \text{height} = \sqrt{27} = 3\sqrt{3} \approx 5.2 \text{ m} \] 3. **Calculate the angle using the cosine function:** \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{6} = \frac{1}{2} \] \[ \theta = \cos^{-1}\left(\frac{1}{2}\right) \] \[ \theta = 60^\circ \] Thus, the angle between the ladder and the ground is \(60^\circ\).