How do you find the position and magnification of a convex mirror?

Since we are given that the height of the object is #0.03# meters high and is placed #0.2# meters away from a convex mirror with focal length #-0.08# meters, we can write out are givens in SI units as:
#d_(obj)=.2#
#h_(obj)=.03# since the image created by convex mirrors are always upright and thus have a positive height value
#f=-0.08# since the focal length of convex mirrors are negative

Consider the following two formulas:

Lensmaker's Formula:
#1/f=1/d_(obj)+1/d_(img)#

Magnification Equation:
#M=h_(img)/h_(obj)=-d_(img)/d_(obj)#

To determine the image's position, we can solve for #d_(img)# in the Lensmaker's Formula with variables only, then plug in the given values to solve:
#1/d_(img)=1/f-1/d_(obj)#
Taking the reciprocal of both sides, we get:
#d_(img)=1/(1/f-1/d_(obj)#

Now, we can substitute the given values of #d_(obj)# and #f# to solve:
#d_(img)=1/(1/-0.08-1/0.2)#
#=-0.05714# #"m"#

Using this value, we can find the magnification of the convex mirror:
#M=-d_(img)/d_(obj)#
#=-(-0.05714)/0.2#
#=0.28571# which has no units

You will need to calculate the distance between the mirror and the image first, which can be done using the mirror equation:

#1/f=1/d_(o)+1/d_i#

where #f# is the focal length, #d_o# is the distance between the mirror and the object, and #d_i# is the distance between the mirror and the image.

We can solve for #d_i#:

#=>1/d_i=1/f-1/d_(o)#

#=>d_i=(1/f-1/d_(o))^-1#

Note that because this is a convex mirror, the focal length must be negative.

Given that #d_o=20.0cm# and #f=-8.00cm# :

#d_i=(-1/8-1/20)^-1#

#=(-7/40)^-1#

#=-40/7cm#

The magnification of a curved mirror can be expressed by the following equation:

#m=-d_i/d_(o)#

Thus we have:

#m=(-(-40/7))/20#

#m=40/140=2/7#

#:.# The position of the image is #40/7# cm behind the mirror and the magnification of the mirror is #2/7#.

This answer makes sense, as a convex mirror will always produce an image which is reduced, upright, and virtual.