To solve the problem, we can use some basic trigonometry. Let's break down the problem step by step.
Given:
- Length of the ladder (hypotenuse, \( c \)) = 20 m
- Distance from the foot of the ladder to the wall (adjacent side, \( a \)) = 9 m
a. Find the angle that the ladder makes with the wall
We can use the cosine function, which is defined as:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Where:
- \(\theta\) is the angle between the ladder and the ground.
- The adjacent side is the distance from the wall (9 m).
- The hypotenuse is the length of the ladder (20 m).
Rearranging the formula, we get:
\[ \theta = \cos^{-1}\left(\frac{9}{20}\right) \]
Calculating this value:
\[ \theta = \cos^{-1}(0.45) \]
Using a calculator, we find:
\[ \theta \approx 63.6^\circ \]
Thus, the angle that the ladder makes with the ground is approximately \( 64^\circ \) to the nearest degree.
b. Find the height above the ground at which the upper end of the ladder touches the wall
We will use the sine function to find the height, defined as:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
Where:
- The opposite side is the height (let's call it \( h \)).
- The hypotenuse is still the length of the ladder (20 m).
Rearranging the formula, we get:
\[ h = \sin(\theta) \cdot c \]
Substituting the known values:
\[ h = \sin(63.6^\circ) \cdot 20 \]
Calculating \( \sin(63.6^\circ) \):
\[ h \approx 0.894 \cdot 20 \approx 17.88 \text{ m} \]
Therefore, the height above the ground at which the upper end of the ladder touches the wall is approximately \( 17.9 \) m (to 1 decimal place).