There are two important kinds of equations that are used when analyzing the effects of curved mirrors...
Mirror Equation: `(1/f) = (1/(di)) + (1/(do))`
"f" represents focal length
"di" represents image distance (the distance between the mirror and image)
"do" represents object distance (the distance between the mirror and the object)
Magnification Equations: `M = ((hi)/(ho))` OR `M = (-(di)/(do))` OR `((hi)/(ho)) = (-(di)/(do))`
"M" represents magnification
"hi" represents image height
"ho" represents object height
Since our problem mentions radius of curvature, it's also helpful to recall the following:
`f xx 2 = c`
"c" represents center of curvature
When solving this type of problem, it's a good idea to start by identifying (1) what information we're given in the problem, (2) what we're trying to find, and (3) an equation that connects everything.
do = 30.0cm
M = +3.0
c = ?
There's only one equation that contains "c", but we don't have enough information to use it yet (because we would need to know "f", and we don't)!
Soooo, can we find "f"? The mirror equation has "f" in it, but we must know "di" and "do" to use it. We're given the value for "do", but since we don't have a value for "di", we don't have enough information to use the mirror equation yet.
Sooooooo, can we find "di"? Well "di" is present in two versions of the magnification equation. Since we don't have information about image height or object height, ignore the versions that include those variables, and use the 2nd option:
`M = (-(di)/(do))`
`(3.0) = (-(di)/(30.0cm))`
`di = -90.0cm`
Now we have enough information to use the mirror equation!
` ` `(1/f) = (1/(di)) + (1/(do))`
`(1/f) = (1/(-90.0cm)) + (1/(30.0cm))`
*remember to take the inverse so that "f" ends up in the numerator
`f = 45.0cm`
Now we have enough information to find "c"!!!
`f xx 2 = c`
`(45.0cm) xx 2 = c`
`c = 90.0cm`