The angle of elevation and depression of top of a statue from a point 32 feet above the lake are respectively `alpha and beta`. If `sinalpha=5/13 and cosbeta=3/5` then what is the height of the statue from the surface of the lake ?

Options

A. 44 feet
B. 42 feet
C. 33 feet
D. 55 feet

Answer

To solve the problem step by step, we will use trigonometric relationships based on the angles of elevation and depression.

### Step 1: Understand the Geometry
We have a point 32 feet above the lake from which the angles of elevation (α) and depression (β) to the top of the statue are given. We need to find the height of the statue (h) from the surface of the lake.

### Step 2: Set Up the Trigonometric Relationships
From the problem, we know:
- \( \sin \alpha = \frac{5}{13} \)
- \( \cos \beta = \frac{3}{5} \)

Using the definitions of sine and cosine:
- For angle α (elevation):
\[ \sin \alpha = \frac{\text{height of statue}}{\text{hypotenuse}} \Rightarrow \sin \alpha = \frac{h}{\sqrt{h^2 + 32^2}} \]
- For angle β (depression):
\[ \cos \beta = \frac{\text{base}}{\text{hypotenuse}} \Rightarrow \cos \beta = \frac{32}{\sqrt{h^2 + 32^2}} \]

### Step 3: Express h in terms of x and y
Let:
- For angle α, we can set \( h = 5x \) and the hypotenuse as \( 13x \).
- For angle β, we can set \( 32 = 3y \) and the hypotenuse as \( 5y \).

### Step 4: Find the Relationships
From the triangle formed by the angle α:
\[ \sqrt{h^2 + 32^2} = 13x \]
Substituting \( h = 5x \):
\[ \sqrt{(5x)^2 + 32^2} = 13x \]
Squaring both sides:
\[ 25x^2 + 1024 = 169x^2 \]
Rearranging gives:
\[ 144x^2 = 1024 \Rightarrow x^2 = \frac{1024}{144} = \frac{64}{9} \Rightarrow x = \frac{8}{3} \]

### Step 5: Calculate h
Now substituting back for h:
\[ h = 5x = 5 \times \frac{8}{3} = \frac{40}{3} \text{ feet} \]

### Step 6: Find y
From the triangle formed by angle β:
\[ \sqrt{h^2 + 32^2} = 5y \]
Substituting \( h = \frac{40}{3} \):
\[ \sqrt{\left(\frac{40}{3}\right)^2 + 32^2} = 5y \]
Squaring both sides:
\[ \frac{1600}{9} + 1024 = 25y^2 \]
Converting 1024 to a fraction with a denominator of 9:
\[ \frac{1600 + 9216}{9} = 25y^2 \Rightarrow \frac{10816}{9} = 25y^2 \]
Thus:
\[ y^2 = \frac{10816}{225} \Rightarrow y = \frac{104}{15} \]

### Final Answer
The height of the statue from the surface of the lake is:
\[ h = \frac{40}{3} \text{ feet} \approx 13.33 \text{ feet} \]