An object 3.28 cm tall is placed 10.5 cm in front of a mirror, which creates an upright image rant is 6.43 cm in height.
A. What is the magnification of the image?
B. What is the radius of the curvature of the mirror?
A. Mag. = 6.43/3.28 = 1.96
B. R = 2 f
Get the focal length f from the equation
1/do + 1/di = 1/f
do = object distance = 10.5 cm
di = image distance = 20.6 cm
How did you get 20.6?
To find the magnification of the image, we use the formula:
magnification = height of image / height of object
Given that the height of the object (Ho) is 3.28 cm and the height of the image (Hi) is 6.43 cm, we can calculate the magnification:
magnification = Hi / Ho
= 6.43 cm / 3.28 cm
= 1.96
Therefore, the magnification of the image is 1.96.
To find the radius of curvature of the mirror, we use the mirror equation:
1 / f = 1 / d0 + 1 / di
Where f is the focal length of the mirror, d0 is the object distance (10.5 cm), and di is the image distance.
Since the mirror creates an upright image, we know that the image distance is negative. Let's assume it to be -di.
Using the given object distance (d0) of 10.5 cm and the magnification (m) of 1.96, we can also use the magnification formula:
magnification = -di / d0
Rearranging the equation, we can solve for di:
-di = m * d0
di = -1.96 * 10.5 cm
di ≈ -20.58 cm
Now, substituting the values into the mirror equation:
1 / f = 1 / 10.5 cm + 1 / (-20.58 cm)
To calculate the radius of curvature (R), we use the formula:
R = 2f
After finding the focal length (f) from the mirror equation, we can calculate the radius of curvature (R).
Solving the mirror equation for f:
1 / f = 1 / 10.5 cm + 1 / (-20.58 cm)
Take the reciprocal of both sides:
f = 1 / (1 / 10.5 cm + 1 / (-20.58 cm))
f ≈ 15.59 cm
Finally, calculating the radius of curvature:
R = 2 * f
R ≈ 2 * 15.59 cm
R ≈ 31.18 cm
Therefore, the radius of curvature of the mirror is approximately 31.18 cm.