To solve this problem, we will apply the lens formula and magnification concept for a convex lens. The given focal length of the lens is \( f = 0.12\,m \). The image is three times the size of the object, which implies a magnification (\( m \)) of \( m = 3 \). Since the image is real, the magnification is negative, so \( m = -3 \).
The magnification formula for a lens is:
\(m = \frac{v}{u}\)
where:
Rearranging the magnification formula gives:
\(v = m \cdot u\)
Substituting \( m = -3 \), we get:
\(v = -3u\)
Now, use the lens formula:
\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)
Substituting the values into the lens formula:
\(\frac{1}{0.12} = \frac{1}{-3u} + \frac{1}{u}\)
Finding a common denominator:
\(\frac{1}{0.12} = \frac{1 - 3}{3u}\)
Simplifying the equation gives:
\(\frac{1}{0.12} = \frac{-2}{3u}\)
This simplifies to:
\(3u \cdot \frac{1}{0.12} = -2\)
Simplify and solve for \( u \):
\(u = \frac{-2 \times 0.12}{3} = -0.08\,m \cdot \frac{1}{3}\)
Simplifying further:
\(u = -0.16 \,m\)
This indicates the object distance is \( 0.16\,m \) from the lens. Since we consider distances in the context of optics, the correct answer with respect to the sign convention used here specifies the positive value relevant to the real scenarios presented in multiple-choice options:
The distance between the object and the lens for a real image is finally \(0.16 \, m\).
Therefore, the correct answer is: