A candle is placed 17.0 cm in front of a convex mirror. When the convex mirror is replaced with a plane mirror, the image moves 6.0 cm farther away from the mirror. Find the focal length of the convex mirror.

Well then, the image distance is 11cm in back of the mirror. Use the lens equation. Watch signs.

-12.36

To find the focal length of the convex mirror, we can use the mirror equation:

1/f = 1/di + 1/do

Where:
- f is the focal length of the mirror
- di is the image distance (distance between the mirror and the image)
- do is the object distance (distance between the mirror and the object)

Given:
- The object distance, do = 17.0 cm
- The change in the image distance, Δdi = 6.0 cm (since the image moves 6.0 cm farther away)

Using the mirror equation, we can rearrange it to solve for the focal length (f):

1/f = 1/di + 1/do

Substituting the values:

1/f = 1/(do + Δdi) + 1/do

Now, we can substitute the given values:

1/f = 1/(17.0 + 6.0) + 1/17.0

Simplifying this equation, we get:

1/f = 1/23.0 + 1/17.0

To add these fractions, we need to find a common denominator, which is equal to the product of the two denominators:

1/f = (17.0 + 23.0) / (17.0 * 23.0) + (23.0 + 17.0) / (17.0 * 23.0)

1/f = 40 / (17.0 * 23.0) + 40 / (17.0 * 23.0)

1/f = (40 + 40) / (17.0 * 23.0)

1/f = 80 / (17.0 * 23.0)

Now, we can simplify further:

1/f ≈ 80 / 391

To find f, we take the reciprocal of both sides of the equation:

f ≈ 391 / 80

Calculating this value using a calculator:

f ≈ 4.89 cm

Therefore, the focal length of the convex mirror is approximately 4.89 cm.

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

1/f = 1/v - 1/u

Where:
f is the focal length
v is the image distance
u is the object distance

We are given that the candle is placed 17.0 cm in front of the convex mirror, so the object distance is u = -17.0 cm (negative because the object is in front of the mirror).
When the convex mirror is replaced with a plane mirror, the image moves 6.0 cm farther away from the mirror. This means that the image distance when the mirror is replaced with a plane mirror is v + 6.0 cm.

Using the mirror formula, we can now substitute the values:

1/f = 1/(v + 6) - 1/(-17)

Simplifying this equation:

1/f = -1/17 - 1/(v + 6)

To get rid of the fractions, we can take the reciprocals:

1/f = -(v + 6)/(17(v + 6)) - 17/(17(v + 6))

Combining the terms:

1/f = (-v - 6 - 17)/(17(v + 6))

Further simplifying:

1/f = (-v - 23)/(17(v + 6))

To isolate f, we can take the reciprocal of both sides:

f = 17(v + 6)/(-v - 23)

So, the focal length of the convex mirror is:

f = 17(v + 6)/(-v - 23) cm