From two points A and B on the same side of a building the angles of elevation of the top of the building are `30^(@) and 60^(@)` respectively. if the height of the building is 10 m find the distances between A and B correct to two decimal places

Options

A. 10 . 66 m
B. 13 . 43 m
C. 11 . 55 m
D. 12 . 26 m

Answer

To solve the problem step by step, we will use trigonometric ratios to find the distances from points A and B to the base of the building.

### Step 1: Understand the setup
We have a building of height \( h = 10 \) m. From point A, the angle of elevation to the top of the building is \( 30^\circ \), and from point B, the angle of elevation is \( 60^\circ \).

### Step 2: Define the distances
Let:
- \( AC \) be the distance from point A to the base of the building (C).
- \( BC \) be the distance from point B to the base of the building (C).
- The distance between points A and B is \( AB = AC + BC \).

### Step 3: Use the tangent function for point B
In triangle \( BDC \) (where D is the top of the building):
\[ \tan(60^\circ) = \frac{h}{BC} \]
Substituting the known values:
\[ \sqrt{3} = \frac{10}{BC} \]
From this, we can solve for \( BC \):
\[ BC = \frac{10}{\sqrt{3}} \approx 5.77 \text{ m} \]

### Step 4: Use the tangent function for point A
In triangle \( ADC \):
\[ \tan(30^\circ) = \frac{h}{AC} \]
Substituting the known values:
\[ \frac{1}{\sqrt{3}} = \frac{10}{AC} \]
From this, we can solve for \( AC \):
\[ AC = 10\sqrt{3} \approx 17.32 \text{ m} \]

### Step 5: Calculate the distance between A and B
Now we can find the distance \( AB \):
\[ AB = AC + BC = 10\sqrt{3} + \frac{10}{\sqrt{3}} \]
To combine these, we can express both terms with a common denominator:
\[ AB = 10\sqrt{3} + \frac{10}{\sqrt{3}} = \frac{10\sqrt{3} \cdot \sqrt{3}}{\sqrt{3}} + \frac{10}{\sqrt{3}} = \frac{30 + 10}{\sqrt{3}} = \frac{40}{\sqrt{3}} \]

### Step 6: Simplify the distance
Now we can simplify \( AB \):
\[ AB \approx \frac{40}{1.732} \approx 23.09 \text{ m} \]

### Step 7: Final answer
Thus, the distance between points A and B is approximately \( 23.09 \) m.