A candle flame of 3 cm is placed at a distance of 3 m from a wall. How far from the wall must a concave mirror be placed in order that it may form an image of flame 9 cm high on th... A candle flame of 3 cm is placed at a distance of 3 m from a wall. How far from the wall must a concave mirror be placed in order that it may form an image of flame 9 cm high on the wall? Read more

Steps to Solve

1. Identify Given Values
   The height of the candle flame ( h_o = 3 , \text{cm} )
   The height of the image ( h_i = 9 , \text{cm} )
   The distance from the candle to the wall ( d_{cw} = 3 , \text{m} = 300 , \text{cm} )

2. Use the Magnification Formula
   The magnification ( m ) of the mirror is given by the formula:
   $$ m = \frac{h_i}{h_o} $$
   Substituting values:
   $$ m = \frac{9 , \text{cm}}{3 , \text{cm}} = 3 $$

3. Relate Object Distance and Image Distance
   The relationship between magnification, object distance ( d_o ), and image distance ( d_i ) is:
   $$ m = -\frac{d_i}{d_o} $$
   Since ( d_o ) is the distance from the candle to the mirror, we have:
   $$ d_o = d_{cw} - d_i $$

4. Set Up the Equation
   Substituting the magnification equation into the relationship:
   $$ 3 = -\frac{d_i}{d_{cw} - d_i} $$
   This rearranges to:
   $$ 3(d_{cw} - d_i) = -d_i $$
   Expanding this gives:
   $$ 3d_{cw} - 3d_i = -d_i $$
   Now, rearranging for ( d_i ):
   $$ 3d_{cw} = 3d_i - d_i $$
   $$ 3d_{cw} = 2d_i $$
   So:
   $$ d_i = \frac{3d_{cw}}{2} $$

5. Calculate Image Distance
   Substituting ( d_{cw} = 300 , \text{cm} ):
   $$ d_i = \frac{3 \cdot 300}{2} = \frac{900}{2} = 450 , \text{cm} $$

6. Calculate Distance from the Wall
   The distance from the mirror to the wall is:
   $$ d_{mw} = d_{cw} - d_i = 300 , \text{cm} - 450 , \text{cm} $$
   Since this value does not make physical sense (it's negative), we conclude that the mirror is placed behind the candle flame, so:
   $$ d_{mw} = d_i - d_{cw} = 450 , \text{cm} - 300 , \text{cm} = 150 , \text{cm} $$

Thus, the mirror must be placed at a distance of ( 450 , \text{cm} ) from the wall to form an image of ( 9 , \text{cm} ) high on the wall.