A high school student studying optics finds that the focal length \(f\) of a lens is related to object and image distances by \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). If \(d_o = 30 \, \text{cm}\) and \(d_i = 60 \, \text{cm}\), what is \(f\) in cm?

["Understanding Lens Formula: Calculating Focal Length from Object and Image Distances", "For high school students studying optics, mastering the lens equation is essential for solving problems involving how lenses form images. The fundamental relationship governing a thin lens is:", "[\n\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\n]", "where:\n- ( f ) is the focal length of the lens (in cm),\n- ( d_o ) is the object distance (in cm) from the lens,\n- ( d_i ) is the image distance (in cm) from the lens.", "This equation connects the physical positions of the object and image to the lens’s defining property—the focal length. Understanding how to apply it helps students predict image formation, magnification, and lens behavior—key skills in physics and applied optics.", "### Applying the Formula: Step-by-Step Example", "Suppose a high school student is analyzing a converging lens setup with:", "- Object distance: ( d_o = 30 , \ ext{cm} )\n- Image distance: ( d_i = 60 , \ ext{cm} )", "We substitute these values into the lens formula:", "[\n\frac{1}{f} = \frac{1}{30} + \frac{1}{60}\n]", "To combine the fractions, find a common denominator:", "[\n\frac{1}{f} = \frac{2}{60} + \frac{1}{60} = \frac{3}{60} = \frac{1}{20}\n]", "Now, solving for ( f ):", "[\nf = 20 , \ ext{cm}\n]", "### Interpretation and Significance", "A positive focal length confirms that the lens is converging, consistent with the given object and image distances. This result illustrates how real-world optical measurements directly relate to theoretical predictions. For students, such problems reinforce algebra and inverse thinking—combining small distances to reveal the lens’s core property.", "Mastering this formula deepens conceptual understanding, enabling students to explore real-world applications like camera lenses, microscopes, and eyeglasses. With practice, calculating focal length becomes intuitive and foundational to further optics study.", "In summary: Given ( d_o = 30 , \ ext{cm} ) and ( d_i = 60 , \ ext{cm} ), the focal length is ( f = 20 , \ ext{cm} )—a clear example of physics in action at the high school level."]