Question

a ladder 20 m long rest against a vertical wall so that the foot of the ladder is 9 m from the wall a find the correct to the nearest degree the angle that the ladder make with the wall b. find correct to the 1. d. p the height above the ground at which the upper and of the ladder touches the wall

Answer 1

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).**