You look into a shaving mirror and the upright image of your face is 40.0 cm away from you. The image has a magnification of 1.50. (a) How far away are you from the mirror? (b) What is the radius of curvature? all answers should be in cm. thanks.
To solve this problem, we can use the mirror equation and magnification formula:
1. The mirror equation is given by:
1/di + 1/do = 1/f
where di is the distance of the image from the mirror (40.0 cm), do is the distance of the object (your face) from the mirror, and f is the focal length of the mirror.
2. The magnification formula is given by:
m = -di/do
where m is the magnification (1.50).
Now, let's solve each part of the problem:
(a) How far away are you from the mirror?
We can rearrange the mirror equation to solve for do:
1/do = 1/f - 1/di
1/do = 1/f - (1/40.0 cm)
1/do = (di - 40.0 cm)/(40.0 cm * di)
We know that di is 40.0 cm and m is 1.50, so we can substitute these values into the magnification formula:
1.50 = -40.0 cm/do
Now, substitute this value of 1/do into the equation derived from the mirror equation:
1/(1.50 * do) = (40.0 cm - 40.0 cm)/(40.0 cm * 40.0 cm)
1/(1.50 * do) = 0 cm^(-1)
To solve for do, we can take the reciprocal of both sides and calculate:
do = 1 / (1.50 * 0 cm^(-1))
do = 0 cm
Therefore, you are 0 cm away from the mirror.
(b) What is the radius of curvature?
Using the mirror equation, we can solve for f:
1/di + 1/do = 1/f
1/40.0 cm + 1/0 cm = 1/f
Since do is 0 cm, the equation becomes:
1/40.0 cm = 1/f
To solve for f, we can take the reciprocal of both sides:
f = 40.0 cm
Therefore, the radius of curvature of the mirror is 40.0 cm.