A convex mirror has a focal length of 19.0 cm. Determine the object location for which the image will be one-fourth as tall as the object.
i really need help on this one. im completely stuck!
Di/Do = -1/4 is the magnification
(The - sign is there because it is a convex mirror with a virtual image.)
1/Do + 1/Di = -1/f = -1/19
1/Do -4/Do = -3/Do = -1/19
Do = 57 cm
Di = -Do/4 = -14.25 cm (behind the mirror)
To determine the object location for which the image formed by a convex mirror will be one-fourth as tall as the object, we can use the mirror equation:
1/f = 1/do + 1/di
where:
f = focal length of the convex mirror
do = object distance from the mirror
di = image distance from the mirror
Given that the focal length (f) of the convex mirror is 19.0 cm, we need to find the object distance (do) for which the image will be one-fourth as tall as the object.
First, we need to recall the magnification equation for a convex mirror:
magnification (m) = -di / do
Since we want the image height (hi) to be one-fourth (1/4) the object height (ho), we can express this as:
hi / ho = -di / do
Given that hi = 1/4 and ho = 1, the equation becomes:
1/4 = -di / do
Now, let's rearrange the equation to solve for -di:
-di = (1/4) * do
Next, let's substitute this value for -di back into the mirror equation:
1/19 = 1/do + 1/[(1/4) * do]
Simplifying this equation will allow us to solve for do, the object distance.
To simplify, let's find the least common denominator:
1/[(1/4) * do] = 4/do
Now, substituting back into the equation:
1/19 = 1/do + 4/do
Combining the fractions:
1/19 = (1 + 4)/do
1/19 = 5/do
Now, to isolate do, let's take the reciprocal of both sides of the equation:
19 = do/5
To solve for do, we can multiply both sides by 5:
95 = do
Therefore, the object distance (do) for which the image formed by the convex mirror will be one-fourth as tall as the object is 95 cm.