A pillar of18 m high. The angles of elevation of the top of a house just in front of it from the top of a pillar, is 30 deg and from the foot is and the house. 45 deg Find the height of the house and the distance between the pillar and the house 

We can solve this problem using trigonometry. Let:

- \( h \) be the height of the house.
- \( d \) be the distance between the pillar and the house.

From the given information:
- The height of the pillar is \( 18 \, \text{m} \).
- The angle of elevation from the top of the pillar to the top of the house is \( 30^\circ \).
- The angle of elevation from the foot of the pillar to the top of the house is \( 45^\circ \).

### Step 1: Use the information from the foot of the pillar (45° angle)

Using the tangent function, since \( \tan(45^\circ) = 1 \):

\[
\tan(45^\circ) = \frac{h}{d}
\]

Since \( \tan(45^\circ) = 1 \), we get:

\[
h = d
\]

### Step 2: Use the information from the top of the pillar (30° angle)

From the top of the pillar, the difference in height is \( (h - 18) \), and the angle of elevation is \( 30^\circ \), so:

\[
\tan(30^\circ) = \frac{h - 18}{d}
\]

Using \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:

\[
\frac{1}{\sqrt{3}} = \frac{h - 18}{d}
\]

Now, substituting \( h = d \) from Step 1 into this equation:

\[
\frac{1}{\sqrt{3}} = \frac{d - 18}{d}
\]

Simplify the equation:

\[
\frac{1}{\sqrt{3}} = 1 - \frac{18}{d}
\]

### Step 3: Solve for \( d \)

Rearrange the equation:

\[
\frac{18}{d} = 1 - \frac{1}{\sqrt{3}}
\]

Simplifying the right-hand side:

\[
1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}}
\]

Now, solving for \( d \):

\[
\frac{18}{d} = \frac{\sqrt{3} - 1}{\sqrt{3}}
\]

\[
d = 18 \times \frac{\sqrt{3}}{\sqrt{3} - 1}
\]

### Step 4: Approximate \( d \)

Approximating \( \sqrt{3} \approx 1.732 \):

\[
d = 18 \times \frac{1.732}{1.732 - 1} = 18 \times \frac{1.732}{0.732}
\]

\[
d \approx 18 \times 2.366 = 42.59 \, \text{m}
\]

So, \( d \approx 42.59 \, \text{m} \).

### Step 5: Find the height \( h \)

Since \( h = d \), the height of the house is approximately:

\[
h \approx 42.59 \, \text{m}
\]

### Final Answer:
- The height of the house is approximately \( 42.59 \, \text{m} \).
- The distance between the pillar and the house is also \( 42.59 \, \text{m} \).

Would you like further details or clarifications?

Here are 5 related questions to explore:
1. How would the problem change if the angle of elevation from the foot was different?
2. Can you solve the problem if only one angle of elevation is given?
3. What is the significance of using trigonometric functions in real-world applications?
4. How does the Pythagorean theorem relate to this problem?
5. How would the calculation change if the pillar was shorter than the house?

**Tip:** In trigonometry problems, drawing a diagram often makes it easier to visualize relationships between angles and sides.

A pillar of 18 m high. The angles of elevation of the top of a house just in front of it from the top of a pillar is 30 degrees, and from the foot of the pillar is 45 degrees. Find the height of the house and the distance between the pillar and the house.

Solve a trigonometry problem involving a pillar and a house. With angles of elevation of 30° and 45°, find the height of the house and the distance between the pillar and the house. Learn to apply tangent functions and trigonometric concepts. Suitable for Grades 10-12.

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