(1) A virtual image -7 cm tall and 17 cm away from a mirror is created by an object 7 cm away from a mirror. How tall is the object? What kind of mirror created the image? What is its focal length?
(2) If a 1.75 m tall man is looking at a plane mirror from an eye which is 15 cm from the top of their head, what is the minimum length that the mirror must be so that he can just barely see his whole body in the mirror? How far up from the ground must the mirror be placed?
duplicate post; already answered
(1) To find the height of the object, we can use the magnification formula:
magnification = height of image / height of object
Given that the height of the virtual image is -7 cm and the object is 7 cm away from the mirror, we can calculate the magnification:
magnification = -7 cm / 7 cm = -1
Since the magnification is negative, it indicates that the image is inverted. Therefore, the object should also be inverted.
To find the type of mirror that created the image, we can use the sign conventions for mirrors. In this case, since both the object distance and image distance are positive, we can conclude that a concave mirror created the image.
Lastly, to find the focal length of the mirror, we can use the mirror equation:
1/focal length = 1/object distance + 1/image distance
Plugging in the values we know:
1/focal length = 1/7 cm + 1/17 cm
Simplifying further leads to:
1/focal length = (17 + 7) / (7 * 17) = 24 / (7 * 17)
Taking the reciprocal of both sides:
focal length = (7 * 17) / 24
Therefore, the focal length of the mirror is approximately 4.83 cm.
(2) To find the minimum length of the mirror required for the man to see his whole body, we need to consider the height of the man's eye and his total height.
Let's denote:
h = height of the man (1.75 m)
e = distance from the top of his head to his eye (15 cm)
m = minimum length of the mirror required
To see his whole body, the line of sight from his eye to the bottom of his feet should just touch the mirror's surface. Therefore, the mirror should have a length equal to the height of the man's eye to cover his entire body.
m = h + e
m = 1.75 m + 0.15 m
m = 1.9 m
So, the minimum length of the mirror must be 1.9 meters.
To determine the height at which the mirror should be placed from the ground, we can consider that the mid-point of the mirror should be at the eye level.
Height of mirror from ground = h/2 + e
Height of mirror from ground = 1.75 m / 2 + 0.15 m
Height of mirror from ground = 0.875 m + 0.15 m
Height of mirror from ground = 1.025 m
Hence, the mirror should be placed approximately 1.025 meters above the ground.