A statue 4 meters high sits on a column 5.6 meters high . How far from the column must a man, whose eye level is 1.6 meters from the ground, stand in order to see the statue at the greatest angle ?

Answer

To solve the problem of how far from the column a man must stand to see the statue at the greatest angle, we can follow these steps:

### Step 1: Understand the Setup
We have a statue that is 4 meters high sitting on a column that is 5.6 meters high. Therefore, the total height of the statue from the ground is:
\[ \text{Total height} = \text{Height of column} + \text{Height of statue} = 5.6 + 4 = 9.6 \text{ meters} \]
The man's eye level is 1.6 meters from the ground.

### Step 2: Define the Variables
Let \( x \) be the distance from the base of the column to the man. The angles we are interested in are:
- \( \alpha \): the angle of elevation from the man's eye level to the top of the statue.
- \( \theta \): the angle of elevation from the man's eye level to the top of the column.

### Step 3: Write the Tangent Functions
Using the definition of tangent for the angles:
- For angle \( \alpha \):
\[ \tan(\alpha) = \frac{\text{Height of statue}}{\text{Distance from man to column}} = \frac{4}{x} \]
- For angle \( \theta \):
\[ \tan(\theta) = \frac{\text{Height of column}}{\text{Distance from man to column}} = \frac{5.6}{x} \]

### Step 4: Use the Angle Addition Formula
To find the angle \( \theta + \alpha \), we can use the tangent addition formula:
\[ \tan(\theta + \alpha) = \frac{\tan(\theta) + \tan(\alpha)}{1 - \tan(\theta) \tan(\alpha)} \]
Substituting the expressions for \( \tan(\theta) \) and \( \tan(\alpha) \):
\[ \tan(\theta + \alpha) = \frac{\frac{5.6}{x} + \frac{4}{x}}{1 - \left(\frac{5.6}{x}\right)\left(\frac{4}{x}\right)} = \frac{\frac{5.6 + 4}{x}}{1 - \frac{22.4}{x^2}} = \frac{\frac{9.6}{x}}{1 - \frac{22.4}{x^2}} \]

### Step 5: Simplify the Expression
This simplifies to:
\[ \tan(\theta + \alpha) = \frac{9.6}{x} \cdot \frac{x^2}{x^2 - 22.4} = \frac{9.6x}{x^2 - 22.4} \]

### Step 6: Maximize the Angle
To find the maximum angle, we need to maximize \( \tan(\theta + \alpha) \). We can differentiate:
\[ y = \frac{9.6x}{x^2 - 22.4} \]
Using the quotient rule:
\[ y' = \frac{(x^2 - 22.4)(9.6) - 9.6x(2x)}{(x^2 - 22.4)^2} = \frac{9.6(x^2 - 22.4 - 2x^2)}{(x^2 - 22.4)^2} = \frac{9.6(-x^2 - 22.4)}{(x^2 - 22.4)^2} \]
Setting the numerator to zero for maximization:
\[ -x^2 - 22.4 = 0 \implies x^2 = 22.4 \implies x = \sqrt{22.4} = 4.73 \text{ meters} \]

### Step 7: Conclusion
The distance from the column at which the man should stand to see the statue at the greatest angle is approximately:
\[ x \approx 4.73 \text{ meters} \]