An obsever looks at a distant tree of height 10m with a telescope of magnifying power of 20. to the observer the tree appears:

Options

A. 10 times nearer.
B. 20 times taller.
C. 20 times nearer.
D. 10 times taller.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the concept of magnifying power
The magnifying power (M) of a telescope is defined as the ratio of the angle subtended by the image (θ_i) to the angle subtended by the object (θ_o). Mathematically, it can be expressed as:
\[ M = \frac{\theta_i}{\theta_o} \]

### Step 2: Relate magnifying power to distances
We know that the angle subtended by an object is related to its height (h) and the distance from the observer (d_o) as:
\[ \theta_o = \frac{h}{d_o} \]
Similarly, for the image:
\[ \theta_i = \frac{h}{d_i} \]
where \( d_i \) is the distance of the image from the observer.

### Step 3: Set up the equation using magnifying power
Substituting these expressions into the magnifying power formula gives us:
\[ M = \frac{\frac{h}{d_i}}{\frac{h}{d_o}} = \frac{d_o}{d_i} \]
From this, we can rearrange to find the relationship between the distances:
\[ d_i = \frac{d_o}{M} \]

### Step 4: Calculate the distances
Given that the height of the tree (h) is 10 m and the magnifying power (M) is 20, we can find the distance of the object (d_o) when the object is at a distance where it appears at its actual height. For practical purposes, we can assume the distance of the object is much larger than the height of the tree, so we can take \( d_o \) as 10 m for simplicity in this context.

Now, substituting the values:
\[ d_i = \frac{10 \, \text{m}}{20} = 0.5 \, \text{m} \]

### Step 5: Interpret the result
The distance \( d_i \) represents how far the image of the tree appears to the observer. Since \( d_i = 0.5 \, \text{m} \), this means that to the observer, the tree appears to be 0.5 m away.

### Conclusion
Thus, to the observer, the tree appears to be 20 times nearer than its actual distance.