A convex spherical mirror has a focal length of -20 cm. An object is placed 30 cm in front of the mirror on the mirror's axis. Where is the image located?
Answer
12 cm in front of the mirror
60 cm behind the mirror
60 cm in front of the mirror
none of the given answers
(30*-20) / (30-(-20)) = -12
Well, this is a humorous mirror indeed! The image formed by this magical mirror is located none of the given answers. In fact, the image formed by a concave spherical mirror with a focal length of -20 cm, when an object is placed 30 cm in front of it, will be 60 cm behind the mirror. So, buckle up and get ready for a wild ride with this mirror!
To determine the location of the image formed by a convex spherical mirror, we can use the mirror formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the mirror
- v is the distance of the image from the mirror
- u is the distance of the object from the mirror
Given:
focal length (f) = -20 cm (negative sign indicates a convex mirror)
distance of the object (u) = -30 cm (negative sign indicates the object is placed in front of the mirror)
Plugging these values into the formula, we get:
1/-20 = 1/v - 1/-30
Simplifying the equation:
-30/-20 = 1/v - (-1/30)
3/2 = 1/v + 1/30
To add fractions, we need a common denominator, which is 30v:
(3/2)(30v) = 30v/v + 30v/30
45v = 30 + v
Moving all terms containing v to one side of the equation:
45v - v = 30
44v = 30
Divide both sides of the equation by 44:
v = 30/44
v ≈ 0.68 cm
The location of the image is 0.68 cm, which is approximately 0.7 cm, in front of the mirror.
Therefore, the correct answer is:
- none of the given answers
To determine the location of the image formed by a convex spherical mirror, we can use the mirror equation:
1/f = 1/d_o + 1/d_i
where:
- f is the focal length of the mirror
- d_o is the object distance (distance of the object from the mirror)
- d_i is the image distance (distance of the image from the mirror)
Given:
- f = -20 cm (the negative sign indicates a convex mirror)
- d_o = 30 cm
Plugging these values into the mirror equation:
1/-20 = 1/30 + 1/d_i
Now, let's solve for d_i:
1/d_i = 1/-20 - 1/30
1/d_i = (-3 - 2)/(-60)
1/d_i = -5/-60
1/d_i = 1/12
Taking the reciprocal of both sides:
d_i = 12 cm
Therefore, the image is located 12 cm in front of the mirror.
So, the correct answer is "12 cm in front of the mirror."