An object is 38.0 cm from a concave mirror of 12 cm focal length. The object is 1.8 cm tall. Use the mirror equation to find the image position.
How high is the image?
To find the image position using the mirror equation, we can use the formula:
1/f = 1/d₀ + 1/dᵢ,
where:
- f is the focal length of the mirror,
- d₀ is the object distance (distance from the object to the mirror), and
- dᵢ is the image distance (distance from the image to the mirror).
Given that the object distance (d₀) is 38.0 cm and the focal length (f) is 12 cm, we can substitute these values into the equation:
1/12 = 1/38 + 1/dᵢ.
To solve for dᵢ, we can rearrange the equation:
1/dᵢ = 1/12 - 1/38.
Now, we find the common denominator and simplify:
1/dᵢ = (38 - 12)/456,
1/dᵢ = 26/456.
Taking the reciprocal of both sides, we get:
dᵢ = 456/26,
dᵢ ≈ 17.54 cm.
So, the image position is approximately 17.54 cm from the concave mirror.
Now, to find the height of the image, we can use the magnification formula given by:
magnification (m) = height of image (hᵢ) / height of object (h₀) = -dᵢ / d₀,
where the negative sign indicates that the image is inverted.
Given that the height of the object (h₀) is 1.8 cm, we can substitute the values of dᵢ and d₀ into the magnification formula:
m = hᵢ / h₀ = -17.54 / 38.
Simplifying, we find:
hᵢ = m * h₀
= -17.54 * 1.8 / 38.
Evaluating this expression, we find:
hᵢ ≈ -0.829 cm.
Therefore, the height of the image is approximately 0.829 cm and it is inverted.