A 13.5​-m fire truck ladder is leaning against a wall. Find the distance d the ladder goes up the wall​ (above the fire​ truck) if the ladder makes an angle of 58 degrees 41 minutes with the horizontal.

To solve this problem, we can use trigonometry. The fire truck ladder forms a right triangle with the wall, where:

- The length of the ladder (hypotenuse) is 13.5 m.
- The angle between the ladder and the horizontal is 58° 41'.

We want to find the distance \( d \), which is the vertical height that the ladder reaches on the wall. This can be found using the sine function from trigonometry:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

Where:
- \( \theta = 58^\circ 41' \) (angle with the horizontal),
- The opposite side is the distance \( d \) we need to find,
- The hypotenuse is the length of the ladder, which is 13.5 m.

We can rearrange the formula to solve for \( d \):

\[
d = \text{hypotenuse} \times \sin(\theta)
\]

### Step 1: Convert the angle into decimal degrees
58° 41' means 58 degrees and 41 minutes. Since there are 60 minutes in a degree, we can convert the minutes into a fraction of a degree:

\[
41' = \frac{41}{60} = 0.6833
\]

So the total angle in decimal degrees is:

\[
\theta = 58 + 0.6833 = 58.6833^\circ
\]

### Step 2: Apply the formula

Now, we use the sine of the angle and the length of the ladder to find \( d \):

\[
d = 13.5 \times \sin(58.6833^\circ)
\]

### Step 3: Calculate

Let's compute this value:

\[
\sin(58.6833^\circ) \approx 0.8541
\]

So,

\[
d \approx 13.5 \times 0.8541 = 11.534 m
\]

Thus, the ladder reaches approximately **11.53 meters** up the wall.

Would you like more details on how the trigonometric functions work, or have any further questions?

### Further Questions:
1. How do you calculate the distance the ladder is from the wall?
2. How can you find the angle if you know the height the ladder reaches and its length?
3. What other trigonometric functions can you use to solve this problem?
4. What would happen if the angle of the ladder changed?
5. How does changing the length of the ladder affect the height it reaches on the wall?

**Tip:** When working with angles in trigonometry, always ensure that your calculator is set to the correct mode (degrees or radians) for accurate results!

Learn how to find the vertical distance a fire truck ladder reaches on a wall using trigonometry. Given a 13.5-meter ladder and an angle of 58° 41', the distance is calculated using the sine function. This problem is suitable for students in Grades 9-12.

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