A ladder 5m long is placed against a wall of a building the foot of the ladder is 2.5m from the base of the wall and in the same horizontal plane. a) Calculate the angle the ladder makes with the wall. b) Calculate the height the ladder reaches up the wall.

Okay, .mliiisah🥷🏾🙌🏾🔥🔥, let's break down this ladder problem.

: a) Calculate the angle the ladder makes with the wall.

Step 1: Identify the sides relative to the angle with the wall. The ladder is the hypotenuse (5 m). The distance from the base of the wall to the foot of the ladder (2.5 m) is the side opposite the angle the ladder makes with the wall. We use the sine function. $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Step 2: Substitute the values and solve for the angle $\theta$. $\sin(\theta) = \frac{2.5\, \text{m}}{5\, \text{m}}$ $\sin(\theta) = 0.5$ To find the angle, we take the inverse sine (arcsin) of 0.5. $\theta = \arcsin(0.5)$ $\theta = \boxed{\text{30}^\circ}$

b) Calculate the height the ladder reaches up the wall.

Step 1: Use the Pythagorean theorem. The ladder, the wall, and the ground form a right-angled triangle. Let $h$ be the height the ladder reaches up the wall. The Pythagorean theorem states: $a^2 + b^2 = c^2$ where $a$ and $b$ are the legs of the triangle, and $c$ is the hypotenuse.

Step 2: Substitute the known values and solve for $h$. The hypotenuse is the ladder length (5 m), and one leg is the distance from the wall (2.5 m). $h^2 + (2.5\, \text{m})^2 = (5\, \text{m})^2$ $h^2 + 6.25\, \text{m}^2 = 25\, \text{m}^2$ $h^2 = 25\, \text{m}^2 - 6.25\, \text{m}^2$ $h^2 = 18.75\, \text{m}^2$ $h = \sqrt{18.75\, \text{m}^2}$ $h \approx \boxed{\text{4.33 m}}$