An object is located 16.8 cm in front of a convex mirror, the image being 11.5 cm behind the mirror. A second object, twice as tall as the first one, is placed in front of the mirror, but at a different location. The image of this second object has the same height as the other image. How far in front of the mirror is the second object located?
I would like to know
To find the distance of the second object from the convex mirror, we can use the mirror formula:
1/f = 1/di + 1/do
where:
f = focal length of the mirror
di = distance of the image from the mirror (negative if it is behind the mirror)
do = distance of the object from the mirror (positive if it is in front of the mirror)
Let's solve the problem step by step:
Step 1: Find the focal length (f)
Since we are given the distances of the object (do) and its corresponding image (di) from the mirror, we can use the mirror formula to find the focal length (f). We can rearrange the formula to solve for f:
1/f = 1/di + 1/do
Substituting the given values, we get:
1/f = 1/-11.5 + 1/16.8
Simplifying this expression gives us:
1/f ≈ -0.08696 + 0.05952
1/f ≈ -0.02744
To find f, we take the reciprocal:
f ≈ -1 / -0.02744
f ≈ 36.4 cm
So, the focal length (f) of the convex mirror is approximately 36.4 cm.
Step 2: Calculate the distance of the second object (do2)
Now that we have the focal length (f), we can use the mirror formula again to find the distance of the second object (do2) from the mirror. We know that the image produced by the second object has the same height as the image produced by the first object, so we can use the magnification formula:
hi/ho = -di/do
where:
hi = height of the image produced by the second object
ho = height of the object
Since the second object is twice as tall as the first one, we have:
hi = 2 * ho
Substituting the values into the magnification formula gives us:
2 * ho / ho = -di/do
Simplifying this expression, we can find the value of -di/do:
-di/do = 2
Now, substituting the values of -di and f into the mirror formula, we get:
1/f = 1/di + 1/do
1/36.4 = -1/11.5 + 1/do2
Next, rearrange the equation to solve for do2:
1/do2 = 1/f - 1/di
1/do2 = 1/36.4 - 1/-11.5
Simplifying this expression gives us:
1/do2 = 0.02747 + 0.08696
1/do2 ≈ 0.11443
To find do2, we take the reciprocal:
do2 ≈ 1/0.11443
do2 ≈ 8.74 cm
Therefore, the second object is located approximately 8.74 cm in front of the convex mirror.