Question

in front of a spherical concave mirror of radius 38.0 cm, you position an object of height 3.44 cm somewhere along the principal axis. the resultant image has a height of 11.35 cm. how far from the mirror is the object located?

Answer 1

Using the mirror formula and magnification formula for a concave mirror, the object distance is calculated based on the given image and object heights. The correct answer depends on this calculation.

To determine the distance from the mirror to the object in front of a concave mirror, we can use the mirror formula:

`\[ (1)/(f) = (1)/(d_o) + (1)/(d_i) \]`

where:

• f is the focal length of the mirror,

• 
  `\( d_o \)` is the object distance,

• 
  `\( d_i \)` is the image distance.

The magnification m is also related to the object and image heights:

`\[ m = -(h_i)/(h_o) \]`

where:

• 
  `\( h_i \)` is the image height,

• 
  `\( h_o \)` is the object height.

For a concave mirror, the focal length is negative. Given that the image is formed on the same side as the object (in front of the mirror), both
`\( d_o \)` and
`\( d_i \)` will be positive.

Given:

• Radius of the mirror R is
  `\( -38.0 \ \text{cm} \)` (negative because it's concave),

• Object height
  `(\( h_o \))` is
  `\( 3.44 \ \text{cm} \)`.

• Image height
  `(\( h_i \))` is
  `\( 11.35 \ \text{cm} \)`.

We can use the mirror formula to find
`\( d_i \)` and then use the magnification formula to find
`\( d_o \)` :

`\[ d_i = (R)/(2) \left(1 + \sqrt{1 + (4h_i^2)/(R^2)}\right) \]`

Once
`\( d_i \)` is determined, we can use the magnification formula to find
`\( d_o \)`:

`\[ d_o = (h_i)/(h_o) \cdot d_i \]`