The image of a crayon appears to be 20.6 cm
behind the surface of a convex mirror and is 3.32 cm tall. The mirror’s focal length is 52.6
cm.
a) How far in front of the mirror is the
crayon positioned?
b) Calculate the magnification of the image
To determine the position and magnification of the image formed by a convex mirror, we can utilize the mirror equation and magnification formula.
a) The mirror equation is given as:
1/f = 1/di + 1/do
where f is the focal length, di is the image distance, and do is the object distance.
We are given the focal length (f = 52.6 cm) and the image distance (di = -20.6 cm) since the image is formed on the opposite side of the mirror. To find the object distance (do), we rearrange the equation:
1/do = 1/f - 1/di
Substituting the given values:
1/do = 1/52.6 - 1/(-20.6)
Simplifying:
1/do = 0.019 - (-0.049)
1/do = 0.068
do = 1 / 0.068
do ≈ 14.7 cm
Therefore, the crayon is positioned approximately 14.7 cm in front of the convex mirror.
b) The magnification (m) can be calculated using the formula:
m = - di / do
Substituting the given values:
m = - (-20.6) / 14.7
m = 1.4
Thus, the magnification of the image is 1.4.
To solve this problem, we can use the mirror equation:
1/f = 1/d_o + 1/d_i
Where:
- f is the focal length of the convex mirror,
- d_o is the object distance (distance of the crayon from the mirror),
- d_i is the image distance (distance of the image from the mirror).
a) To find the object distance, we rearrange the equation:
1/d_o = 1/f - 1/d_i
Substituting the given values:
f = 52.6 cm
d_i = -20.6 cm (negative sign indicates the image is behind the mirror)
1/d_o = 1/52.6 - 1/-20.6
Simplifying the equation:
1/d_o = 0.01902 cm^(-1)
Now, we can find the value of d_o by taking the reciprocal:
d_o = 1/0.01902 cm^(-1)
d_o ≈ 52.55 cm
Therefore, the crayon is positioned approximately 52.55 cm in front of the mirror.
b) The magnification of the image, denoted as "m", is given by the formula:
m = -d_i / d_o
Substituting the given values:
m = -(-20.6 cm) / 52.55 cm
m ≈ 0.3928
Therefore, the magnification of the image is approximately 0.3928.