Final answer:
The object distance for a concave mirror with a focal length of 48.0 cm, where the image distance is two-thirds of the object distance, is 72.0 cm.
Step-by-step explanation:
To determine the object distance for a concave mirror with a given focal length and the condition that the image distance is two-thirds the object distance, we can use the mirror equation.
The mirror equation is given by:
\(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\)
Where:
- f is the focal length of the mirror
- d_o is the object distance from the mirror
- d_i is the image distance from the mirror
Given that f = 48.0 cm and d_i = \frac{2}{3}d_o, we substitute the second expression into the mirror equation and solve for d_o.
\(\frac{1}{48} = \frac{1}{d_o} + \frac{3}{2d_o}\)
Combining terms and solving for d_o gives us:
d_o = 72 cm. Thus, the object distance is 72.0 cm.