A 6.0 cm tall object is placed 20 cm in front of a convex mirror with focal length −100 cm. What is the size of the image formed?
A) 5.0 cm
B) 7.2 cm
C) 7.5 cm
D) 30 cm
E) 12 cm

Question

Quizplus · Accepted Answer

The size of the image formed by a convex mirror can be calculated using the mirror equation and the magnification equation. The mirror equation is $ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $, where $ f $ is the focal length, $ d_o $ is the object distance, and $ d_i $ is the image distance. The magnification $ m $ is given by $ m = \frac{h_i}{h_o} = \frac{-d_i}{d_o} $, where $ h_i $ is the image height and $ h_o $ is the object height.Given: $ f = -100 $ cm (convex mirror, hence negative), $ d_o = 20 $ cm, $ h_o = 6.0 $ cm.First, find $ d_i $ using the mirror equation:$$ \frac{1}{-100} = \frac{1}{20} + \frac{1}{d_i} $$$$ \frac{1}{d_i} = \frac{1}{-100} - \frac{1}{20} $$$$ \frac{1}{d_i} = -\frac{1}{25} $$$$ d_i = -25 $$ cm (negative indicating that the image is virtual and formed on the same side as the object).Next, find $ h_i $ using the magnification equation:$$ m = \frac{h_i}{6.0} = \frac{-(-25)}{20} $$$$ m = \frac{25}{20} = 1.25 $$$$ h_i = 1.25 \times 6.0 = 7.5 $$ cm.However, based on the calculation, it seems there was an error in my explanation regarding the correct choice. The correct calculation should lead to the correct answer based on the given options and the calculations provided. Let's correct the calculation mistake:Given that the object distance $ d_o = 20 $ cm and the focal length $ f = -100 $ cm, we first find the image distance $ d_i $ using the mirror equation:$$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$$$ \frac{1}{-100} = \frac{1}{20} + \frac{1}{d_i} $$$$ \frac{1}{d_i} = \frac{1}{-100} - \frac{1}{20} $$$$ \frac{1}{d_i} = -\frac{1}{25} $$$$ d_i = -25 $$ cm.Then, using the magnification equation to find the image height $ h_i $:$$ m = \frac{h_i}{h_o} = \frac{-d_i}{d_o} $$$$ m = \frac{-(-25)}{20} = \frac{25}{20} = 1.25 $$$$ h_i = m \times h_o = 1.25 \times 6.0 = 7.5 $$ cm.Given the mistake in my initial explanation and the correction provided, the correct answer based on the corrected calculation should indeed be:C) 7.5 cm.

A 60 Cm Tall Object Is Placed 20 Cm in Front