A candle flame is placed in front of a concave mirror on the principal axis such that is twice the size of the object and is inverted. If the sum of the object distance and the image distance is 72 cm, how far from the mirror is the image and what is the focal length of the mirror.
You left out some words after "that" in the second line. You are probably talking about the image there.
Use the lens equation and do the algebra
To solve this problem, we can make use of two properties of concave mirrors: the mirror equation and the magnification formula.
Let's denote the object distance as "u," the image distance as "v," and the focal length as "f."
1. The mirror equation:
The mirror equation is given by:
1/v + 1/u = 1/f
In this case, the given information is that the sum of the object distance and the image distance is 72 cm
So, we have:
u + v = 72 ----(1)
2. The magnification formula:
The magnification formula is given by:
magnification (m) = (height of the image) / (height of the object)
Since it is mentioned that the size of the image is twice the size of the object and is inverted, the magnification would be -2.
So, we have:
m = -2
Now, let's solve the problem step by step:
Step 1: Determine the relationship between object distance and image distance.
Since the image is inverted, the magnification is negative, which implies that the image distance is negative. So, we have:
v = -2u
Step 2: Substitute the value of 'v' obtained from the relationship into the mirror equation (equation 1):
u + (-2u) = 72
Simplifying the above equation:
-u = 72
u = -72
Step 3: Substitute the value of 'u' back into the relationship between object and image distance:
v = -2(-72)
v = 144
Step 4: Determine the focal length:
To find the focal length, we can use the mirror equation. Substitute the values of 'v' and 'u' into the mirror equation:
1/v + 1/u = 1/f
1/144 + 1/-72 = 1/f
Simplifying the above equation:
1/f = 1/144 - 1/72
1/f = (1 - 2)/144
1/f = -1/144
Therefore, the focal length of the mirror is -144 cm (negative sign indicates that it is a concave mirror).
Step 5: Determine the distance from the mirror to the image:
The distance from the mirror to the image is given by the absolute value of the image distance:
Distance = |v| = |144| = 144 cm
Therefore, the image is 144 cm from the mirror, and the focal length of the mirror is -144 cm (or 144 cm in magnitude).