Final answer:
The object distance is 7.5 cm when a lens produces an image that is twice the size of the object, with the image located 15 cm away from the lens.
Step-by-step explanation:
To determine the object distance when a lens produces an image twice as large as the object and the image is located 15 cm from the lens, we can use the lens formula:
\(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\)
Where:
- f is the focal length of the lens,
- do is the object distance,
- di is the image distance, which is given as 15 cm.
Because the image is twice the size of the object, the magnification (m) is -2. We take the negative sign because a real image, which is inverted when compared to the object, has a negative magnification. The magnification is also the ratio of the image distance to the object distance: m = -\(\frac{d_i}{d_o}\).
Solving for the object distance using the magnification, we get:
\(d_o = -\frac{d_i}{m} = -\frac{15cm}{-2} = 7.5cm\)
So the object distance is 7.5 cm.