A convex surveillance mirror in a convenience store has a focal length of 10.40 m A customer, who is 1.7 all, is standing 4.5 m ir front of the mirror a. Calculate the image distance. b. Calculate the image height

To solve this problem, we will use the mirror equation and magnification formula. The mirror equation is given by:<br /><br />\[<br />\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}<br />\]<br /><br />where:<br />- \( f \) is the focal length of the mirror,<br />- \( d_o \) is the object distance (distance from the object to the mirror),<br />- \( d_i \) is the image distance (distance from the image to the mirror).<br /><br />The magnification \( m \) is given by:<br /><br />\[<br />m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}<br />\]<br /><br />where:<br />- \( h_i \) is the image height,<br />- \( h_o \) is the object height.<br /><br />Given:<br />- Focal length \( f = -0.40 \, \text{m} \) (negative for convex mirrors),<br />- Object distance \( d_o = 4.5 \, \text{m} \),<br />- Object height \( h_o = 1.7 \, \text{m} \).<br /><br />### a. Calculate the image distance (\( d_i \))<br /><br />Using the mirror equation:<br /><br />\[<br />\frac{1}{-0.40} = \frac{1}{4.5} + \frac{1}{d_i}<br />\]<br /><br />First, calculate \(\frac{1}{4.5}\):<br /><br />\[<br />\frac{1}{4.5} \approx 0.2222<br />\]<br /><br />Now substitute into the equation:<br /><br />\[<br />-\frac{1}{0.40} = 0.2222 + \frac{1}{d_i}<br />\]<br /><br />\[<br />-2.5 = 0.2222 + \frac{1}{d_i}<br />\]<br /><br />Subtract 0.2222 from both sides:<br /><br />\[<br />-2.7222 = \frac{1}{d_i}<br />\]<br /><br />Now take the reciprocal to find \( d_i \):<br /><br />\[<br />d_i \approx -\frac{1}{2.7222} \approx -0.3673 \, \text{m}<br />\]<br /><br />The negative sign indicates that the image is virtual and located on the same side of the mirror as the object.<br /><br />### b. Calculate the image height (\( h_i \))<br /><br />Using the magnification formula:<br /><br />\[<br />m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}<br />\]<br /><br />Substitute the known values:<br /><br />\[<br />m = -\frac{-0.3673}{4.5} \approx 0.0816<br />\]<br /><br />Now calculate the image height:<br /><br />\[<br />h_i = m \times h_o = 0.0816 \times 1.7 \approx 0.1387 \, \text{m}<br />\]<br /><br />Thus, the image height is approximately \( 0.1387 \, \text{m} \).