Question

An unknown lens forms an image of an object that is \(24 \mathrm{~cm}\) away from the lens, inverted, and a factor of 4 larger in size than the object. Where is the object located? a) \(6 \mathrm{~cm}\) from the lens on the same side of the lens b) \(6 \mathrm{~cm}\) from the lens on the other side of the lens c) \(96 \mathrm{~cm}\) from the lens on the same side of the lens d) \(96 \mathrm{~cm}\) from the lens on the other side of the lens e) No object could have formed this image.

Short answer

Short answer: The object is located 6 cm from the lens on the other side of the lens.

Step-by-step solution

Write down the lens equation and the magnification equation.

The lens equation is given by: \begin{align} \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \end{align} where \(f\) is the focal length of the lens, \(d_o\) is the object distance, and \(d_i\) is the image distance. The magnification equation is given by: \begin{align} M = -\frac{d_i}{d_o} \end{align} where \(M\) is the magnification. In this problem, we have \(d_i = 24\mathrm{~cm}\) and \(M = 4\).

Solve for the object distance using the magnification equation.

Since the image is inverted and 4 times larger, we have \(M = -4\). Using the magnification equation: \begin{align} -4 = -\frac{d_i}{d_o} \end{align} Cross-multiplying the equation, we get: \begin{align} 4d_o = d_i \end{align} We know \(d_i = 24\mathrm{~cm}\), so we can substitute to find \(d_o\): \begin{align} 4d_o = 24 \Rightarrow d_o = 6\mathrm{~cm} \end{align}

Determine the correct answer regarding the object's location.

We found that the object is \(6\mathrm{~cm}\) away from the lens. Since the image is inverted and magnified, it must be a real image formed on the other side of the lens by a converging lens. Therefore, the correct answer is: b) \(6 \mathrm{~cm}\) from the lens on the other side of the lens.