An object that is 32 cm in front of a convex mirror has an image located 17 cm behind the mirror. How far behind the mirror is the image located when the object is 24 cm in front of the mirror?
To find the distance behind the mirror at which the image is located when the object is 24 cm in front of the mirror, we can use the mirror formula:
1/f = 1/d_o + 1/d_i
Where:
f is the focal length of the mirror,
d_o is the distance of the object from the mirror, and
d_i is the distance of the image from the mirror.
Given that the object is 32 cm in front of the mirror and the image is located 17 cm behind the mirror, we can substitute these values into the formula:
1/f = 1/32 + 1/-17
To simplify this equation, we can calculate the reciprocal of each term:
1/f = 1/32 - 1/17
To combine these fractions, we need to find a common denominator. The least common multiple (LCM) of 32 and 17 is 544, so we can rewrite the equation as:
1/f = (17 - 32)/544
Simplifying, we get:
1/f = -15/544
To isolate f, the equation can be rewritten as:
f = -544/15
Now that we have found the value of the focal length (f), we can use it to find the distance of the image when the object is 24 cm in front of the mirror. Since the image is formed by a convex mirror, it is virtual, meaning it is located on the same side as the object.
Using the mirror formula again, we can substitute the given values:
1/f = 1/d_o + 1/d_i
1/-544/15 = 1/24 + 1/d_i
Simplifying, we get:
-15/544 = 1/24 + 1/d_i
To combine these fractions, we need to find a common denominator again. The least common multiple (LCM) of 24 and 544 is 13,056. Rewriting the equation with this common denominator, we get:
-15/544 = (544 + 13,056)/d_i
Simplifying, we get:
-15/544 = 13,600/d_i
To isolate d_i, we can cross multiply:
(-15)(d_i) = (544)(13,600)
Solving for d_i, we find:
d_i = (544)(13,600)/-15
Calculating this expression, we get:
d_i ≈ -496,000 cm
Therefore, the image is located approximately 496,000 cm (or 4960 m) behind the mirror when the object is 24 cm in front of the mirror.