A 3cm tall object is placed 40cm from a concave mirror with a focal distance of 16cm. The object is moved to 10cm from the same mirror. How would you calculate the distance to the image from the lens, the magnification, and the height of the image?

Question

Adrian Ayala · Accepted Answer

#d_i=-80/3cm#

#m=8/3#

#h_i=8cm# "Lens" should, in my opinion, be "mirror." To determine the image distance, we can apply the mirror equation: #1/f=1/d_i+1/d_o# Where #f# is the focal length, #d_i# is the distance from the mirror to the image, and #d_o# is the distance from the mirror to the object. We can then use our values for #d_i# and #d_o# to calculate the magnification, which is given by the equation: #m=(-d_i)/(d_o)# Additionally, making use of the fact that #m=(-d_i)/(d_o)=h_i/h_o# we can solve for #h_i# and use the value we obtain for #m# as well as #h_o# to calculate: #h_i=m*h_o# It is provided to us that: #h_o=3cm# #f=16cm# #d_o=10cm# #1.# Calculate #d_i#: #1/f=1/d_i+1/d_o# #=>1/d_i=1/f-1/d_o# #=>d_i=(1/f-1/d_o)^-1# #=(1/16-1/10)^-1# #=-80/3cm# #~~ -26.7cm# #2.# Calculate #m#: #m=-d_i/d_o# #=-((-80/3))/10# #=80/30# #=8/3# #3.# Calculate #h_i#: #m=h_i/h_o# #=>h_i=m*h_o# #=(8/3)(3)# #=8cm#

Emily Abbate · Answer

undefined To calculate the distance to the image from the lens (image distance), the magnification, and the height of the image, you can use the mirror equation and the magnification formula.

1. **Distance to the image from the lens (image distance)**:
$$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$
Where:
   - $ f $ = focal length of the concave mirror (given as -16 cm since it's concave)
   - $ d_o $ = object distance (initially 40 cm, then moved to 10 cm)
   - $ d_i $ = image distance (unknown)

2. **Magnification (height of the image)**:
$$ M = -\frac{d_i}{d_o} $$
Where:
   - $ M $ = magnification
   - $ d_i $ = image distance (from the first calculation)
   - $ d_o $ = object distance (initially 40 cm, then moved to 10 cm)

3. **Height of the image**:
$$ h_i = M \times h_o $$
Where:
   - $ h_i $ = height of the image
   - $ M $ = magnification (from the second calculation)
   - $ h_o $ = height of the object (given as 3 cm)