The magnification of a convex mirror is +0.65X for objects 2.8 m from the mirror. What is the focal length of this mirror?

Well, let's see, with a magnification of +0.65X, it seems like this convex mirror will make everything look smaller, but still the right way up. As for the focal length, I'm not sure. Maybe the mirror likes to keep its focal length a closely guarded secret. It's like the mirror is saying, "I'm not going to focus on one thing, I'll focus on everything!" Talk about spreading the love! So, sorry, I don't have the focal length for you, but at least we have a mirror that's a master of inclusivity!

To find the focal length of a convex mirror, we can use the mirror formula, which is:

1/f = 1/v - 1/u

Where:
f = focal length of the mirror
v = image distance
u = object distance

Given the magnification (M) of +0.65X, we can use the formula:

M = -v/u

Given that M = +0.65X, we can rewrite it as:

0.65 = -v/u

Since a convex mirror always produces a virtual image, we know that the image distance (v) is negative. So, let's substitute v = -2.8 m into the magnification equation:

0.65 = -(-2.8)/u
0.65 = 2.8/u

Now, let's solve for u:

u = 2.8/0.65
u ≈ 4.31 m

Now, we can substitute the values of u and v into the mirror formula to find the focal length (f):

1/f = 1/v - 1/u
1/f = 1/(-2.8) - 1/(+4.31)

Simplifying further:

1/f = -0.3571 - 0.2322
1/f ≈ -0.5893

Now, we can solve for the focal length (f):

f = 1/(-0.5893)
f ≈ -1.697 m

Therefore, the focal length of this convex mirror is approximately -1.697 meters.

To find the focal length of a convex mirror using the given magnification and the distance of an object from the mirror, we can use the mirror formula:

1/f = 1/v - 1/u

Where:
f = focal length of the mirror
v = image distance from the mirror
u = object distance from the mirror

Given:
Magnification (m) = +0.65X (positive indicates an upright virtual image)
Object distance (u) = 2.8 m

First, let's express the magnification in terms of the image and object distances:

m = -v/u (for convex mirrors)

Rearranging the equation, we can solve for the image distance (v):

v = -m * u

Plugging in the given values, we get:

v = -0.65 * 2.8
v ≈ -1.82 m

Since the image formed by a convex mirror is virtual and upright, the image distance (v) is negative. The negative sign indicates that the image is formed behind the mirror.

Now, substituting the values for v and u into the mirror formula, we can solve for the focal length (f):

1/f = 1/v - 1/u

1/f = 1/(-1.82) - 1/2.8
1/f ≈ -0.5495 + 0.3571

1/f ≈ -0.1924

f ≈ 1/(-0.1924)
f ≈ -5.195 m

The focal length of the convex mirror is approximately -5.195 m. Note that the negative sign indicates that the focal length is behind the mirror.

Since it is convex, the focal length is negative. From the magnification,you know that the image distance is Di = -0.65 Do. The image distance Di is negative because it is a virtual image, behind the mirror.

Now solve

1/Do + 1/Di = 1/Do -1/0.65 Do = 1/f
(1 - 1.538)/Do = 1/f

-0.538/Do = 1/f

f = -1.86 Do = -5.2 m

Thank you!