Identical objects are located at the same distance from two spherical mirrors, A and B. The magnifications produced by the mirrors are mA = 7.2 and mB = 1.1. Find the ratio fA/fB of the focal lengths of the mirrors.
Well, it seems like mirrors A and B are quite the characters! One of them is really good at magnifying things, with a magnification of 7.2, while the other is just average with a magnification of 1.1. Poor Mirror B must feel a bit overshadowed!
Now, let's find the ratio of their focal lengths, fA and fB. We can use the magnification formula for spherical mirrors, which is given by:
magnification = -f / u
Here, fA and fB are the focal lengths of the mirrors, and u is the object distance.
For Mirror A, the magnification is 7.2. So we have:
7.2 = -fA / u
Similarly, for Mirror B, the magnification is 1.1:
1.1 = -fB / u
Since the objects are identical and located at the same distance from both mirrors, the object distance u is the same for both mirrors.
So we can rearrange the equations to solve for the ratio fA/fB:
7.2 = -fA / u ----- Equation 1
1.1 = -fB / u ----- Equation 2
Dividing Equation 1 by Equation 2 gives us:
7.2/1.1 = -fA / (-fB)
Simplifying further:
6.545 = fA/fB
Now, isn't that a funny ratio? It's almost like the focal length of Mirror A is doing a little jig while Mirror B is just average, trying to keep up! The ratio fA/fB is approximately 6.545.
I hope that answers your question! And remember, mirrors can always reflect their own quirks and personalities, just like people!
To find the ratio fA/fB of the focal lengths of the mirrors, we need to use the magnification formula for spherical mirrors.
The magnification formula for a spherical mirror is given by:
m = - (image distance)/(object distance)
Given that the magnifications produced by mirrors A and B are mA = 7.2 and mB = 1.1 respectively, we can set up the following equations:
mA = -dA/do
mB = -dB/do
where dA and dB are the image distances and do is the object distance.
Since the objects are located at the same distance from both mirrors, the object distance (do) will be the same for both mirrors. Thus, we can write:
dA/do = dB/do
Multiplying both sides by do, we get:
dA = dB
Now let's solve for the focal lengths of the mirrors using the mirror formula:
1/f = 1/do + 1/di
where f is the focal length, do is the object distance, and di is the image distance.
For mirror A, we have:
1/fA = 1/do + 1/dA
For mirror B, we have:
1/fB = 1/do + 1/dB
Substituting dA = dB, we get:
1/fA = 1/do + 1/dB
1/fB = 1/do + 1/dB
Now we can find the ratio fA/fB by dividing the equation for fA by the equation for fB:
(fA/fB) = (1/fA) / (1/fB)
= [(1/do + 1/dB)/(1/do + 1/dB)]
= (1/do + 1/dB) * (1/do + 1/dB)
Since dA = dB and we need to find the ratio of the focal lengths, the ratio fA/fB simplifies to:
(fA/fB) = (1/do + 1/dA) * (1/do + 1/dA)
Now we can substitute the given magnifications into our equation for magnification:
mA = -dA/do
7.2 = -dA/do
dA = -do/7.2
mB = -dB/do
1.1 = -dA/do
dA = -do/1.1
Since dA = dB, we can set the two equations equal to each other:
-do/7.2 = -do/1.1
Cross-multiplying and simplifying, we get:
1.1 * (-do) = 7.2 * (-do)
Simplifying further, we find:
-1.1do = -7.2do
Dividing both sides by do, we get:
-1.1 = -7.2
This equation is not possible, which implies that there is an error in the given problem statement.
To find the ratio of the focal lengths of the mirrors, we need to use the mirror formula and the magnification formula.
The mirror formula relates the object distance (u), the image distance (v), and the focal length (f) of a spherical mirror:
1/f = 1/v + 1/u
The magnification formula relates the magnification (m), the image distance (v), and the object distance (u):
m = -v/u
Since the identical objects are located at the same distance from both mirrors, the object distance (u) will be the same for both mirrors.
Let's denote the focal length of mirror A as fA and the focal length of mirror B as fB. Given that the magnification produced by mirror A is mA = 7.2 and the magnification produced by mirror B is mB = 1.1, we can set up the following equations:
mA = -vA/u = -fA/u
mB = -vB/u = -fB/u
Rearranging these equations gives us:
fA/u = -mA
fB/u = -mB
Now, we can find the ratio of the focal lengths by dividing the equation for fA/u by the equation for fB/u:
(fA/u) / (fB/u) = (-mA) / (-mB)
fA / fB = mA / mB
Substituting the given values:
fA / fB = 7.2 / 1.1
Therefore, the ratio of the focal lengths of the mirrors is approximately 6.546:1.