Question

The passenger side of your car has a side-view mirror with a convex sphericalmirror. When you pass a car, you see its reflection. If the image is 9.0 cm tall andthe car is 1.5 m tall, what is the mirror’s magnification? If the car is 3 m from themirror, what is the focal length of the mirror? What is the mirror’s radius ofcurvature?

Answer 1

The mirror’s magnification is 0.06

The focal length is equal to 19.15 cm.

The mirror's radius of curvature is 38.3 cm

Explanation:

Given:

height image = 9 cm

height object = 1.5 m = 150 cm

The mirror's magnification is calculated using the following formula:

`m=(h_i)/(h_o)=(9)/(150)=0.06`

The mirror’s magnification is 0.06

Since we know the distance of the object which is -3 m (-300 cm) and the mirror's magnification we can calculate the distance of the image, just like this:

`\begin{gathered} m=(-d_i)/(d_o) \\ \\ 0.06=(-d_i)/(-300) \\ \\ d_i=300\cdot0.06 \\ \\ d_i=18\text{ cm} \end{gathered}`

We calculate the focal length using the following formula:

`\begin{gathered} (1)/(f)=(1)/(d_i)+(1)/(d_o) \\ \\ (1)/(f)=(1)/(18)+(1)/(-300) \\ \\ (1)/(f)=(18-300)/(-5400) \\ \\ f=(-5400)/(-282) \\ \\ f=\:19.15\text{ cm} \end{gathered}`

The focal length is equal to 19.15 cm.

Finally we calculate the mirror's radius of curvature knowing that twice the focal length, therefore:

`\begin{gathered} r=2f=19.15\cdot2 \\ \\ r=38.3\text{ cm} \end{gathered}`

The mirror's radius of curvature is 38.3 cm