building on a slope at 16 degrees to the horizontal (AD), From point A, 2 metres down the slope, a 7 metre ladder is placed against the wall, use triangle ACD to find AD. give your answer in 2 decimal place, hence or otherwise find BAC

To solve this problem, we'll start by analyzing the given information and using trigonometric relationships to find the lengths and angles involved.

### Step 1: Analyze the problem and diagram

- The building is on a slope inclined at 16 degrees to the horizontal.
- From point \( A \), 2 meters down the slope is point \( D \).
- A 7-meter ladder is placed from point \( D \) against the wall, reaching point \( C \).

We need to find the length of \( AD \) and the angle \( BAC \).

Let's draw a diagram to visualize this:
```
C
|
|\
| \
|  \
|   \
A----D
```

- \( AD \) is the horizontal distance down the slope.
- \( DC \) is the length of the ladder (7 meters).
- \( \angle DAB = 16^\circ \) (slope angle).

### Step 2: Find \( AD \)

Using trigonometric functions in right triangle \( ABD \):

In \( \triangle ABD \):
- \( AB \) is the vertical distance from \( A \) down the slope (which is 2 meters vertically).
- \( AD \) is the horizontal component we need to find.

From trigonometry, we know:
\[
AD = \frac{AB}{\sin(16^\circ)}
\]

Given \( AB = 2 \) meters:
\[
AD = \frac{2}{\sin(16^\circ)}
\]

Calculate \( AD \):
\[
\sin(16^\circ) \approx 0.2756
\]
\[
AD = \frac{2}{0.2756} \approx 7.26 \, \text{meters}
\]

### Step 3: Find angle \( BAC \)

In \( \triangle ACD \):
- \( AC \) is the hypotenuse of the right triangle formed by the ladder.
- \( AD \) is the base (7.26 meters).
- \( CD \) is the length of the ladder (7 meters).

Using the Pythagorean theorem, we can first find \( AC \):
\[
AC^2 = AD^2 + CD^2
\]
\[
AC^2 = (7.26)^2 + 7^2
\]
\[
AC^2 = 52.7076 + 49
\]
\[
AC^2 = 101.7076
\]
\[
AC = \sqrt{101.7076} \approx 10.08 \, \text{meters}
\]

Now, use the cosine rule to find \( \angle BAC \):
\[
\cos(\angle BAC) = \frac{AD}{AC}
\]
\[
\cos(\angle BAC) = \frac{7.26}{10.08} \approx 0.7206
\]
\[
\angle BAC = \cos^{-1}(0.7206) \approx 43.94^\circ
\]

### Summary

1. The length \( AD \) is approximately \( 7.26 \) meters (to 2 decimal places).
2. The angle \( \angle BAC \) is approximately \( 43.94^\circ \).

Would you like more details on any part of this solution, or do you have any questions?

### Related Questions:

1. How would you find the height of point \( C \) above point \( A \)?
2. Can you explain how to use the Pythagorean theorem in inclined plane problems?
3. What is the significance of the sine function in solving inclined plane problems?
4. How do you calculate angles using the cosine rule?
5. What are the implications of a steeper slope on the length \( AD \)?
6. How would the problem change if the ladder were 8 meters instead of 7 meters?
7. Can you explain the concept of angles of elevation and depression in this context?
8. How do you verify the accuracy of trigonometric calculations in real-world problems?

### Tip:

When solving problems involving slopes and angles, always draw a clear diagram to visualize the scenario. This helps in identifying the right trigonometric relationships and applying them correctly.

Learn how to solve the triangle problem on a 16 degree slope, involving the calculation of AD and angle BAC. This problem uses trigonometric relationships such as the sine function and cosine rule to find lengths and angles. Suitable for high school students.

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