The angles of elevation of the top of a temple , from the foot and the top of a building 30 m high are `60^(@)and30^(@)` respectively .Then the height of the temple is

Options

A. 50 metre
B. 43 metre
C. 40 metre
D. 45 metre

Answer

To find the height of the temple given the angles of elevation from the foot and the top of a building, we can follow these steps:

### Step 1: Understand the Problem
We have a building that is 30 meters high. From the foot of the building, the angle of elevation to the top of the temple is 60 degrees. From the top of the building, the angle of elevation to the top of the temple is 30 degrees. We need to find the height of the temple.

### Step 2: Set Up the Diagram
- Let the height of the temple be \( h \).
- The height of the building is 30 meters.
- Let \( d \) be the horizontal distance from the foot of the building to the base of the temple.

### Step 3: Use Trigonometric Ratios
1. From the foot of the building (30 m height) to the top of the temple (height \( h \)):
- The angle of elevation is 60 degrees.
- We can use the tangent function:
\[ \tan(60^\circ) = \frac{h - 30}{d} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{h - 30}{d} \quad \text{(Equation 1)} \]

2. From the top of the building (30 m height) to the top of the temple (height \( h \)):
- The angle of elevation is 30 degrees.
- Again, using the tangent function:
\[ \tan(30^\circ) = \frac{h - 30}{d} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{d} \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
From Equation 2, we can express \( d \) in terms of \( h \):
\[ d = h \sqrt{3} \]

Now substitute \( d \) in Equation 1:
\[ \sqrt{3} = \frac{h - 30}{h \sqrt{3}} \]
Cross-multiplying gives:
\[ \sqrt{3} \cdot h \sqrt{3} = h - 30 \]
This simplifies to:
\[ 3h = h - 30 \]
Rearranging gives:
\[ 3h - h = -30 \]
\[ 2h = 30 \]
\[ h = 15 \]

### Step 5: Calculate the Total Height of the Temple
The total height of the temple is the height of the building plus the height \( h \):
\[ \text{Total height of the temple} = 30 + h = 30 + 15 = 45 \text{ meters} \]

### Final Answer
The height of the temple is **45 meters**.