a 3.0 cm object is 12.0 cm from a convex mirror that has a focal length of 20.0 cm. what is the distance of…

Explanation:

Step1: Use mirror formula

The mirror formula 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. For a convex mirror, $f=- 20.0\ cm$ and $d_o = 12.0\ cm$. Substitute the values into the formula: $\frac{1}{-20}=\frac{1}{12}+\frac{1}{d_i}$.

Step2: Solve for $d_i$

First, rewrite the equation to isolate $\frac{1}{d_i}$: $\frac{1}{d_i}=\frac{1}{-20}-\frac{1}{12}$. Find a common denominator: $\frac{1}{d_i}=\frac{-3 - 5}{60}=\frac{-8}{60}=\frac{-2}{15}$. Then, $d_i=-7.5\ cm$.

Step3: Use magnification formula

The magnification formula is $m =-\frac{d_i}{d_o}=\frac{h_i}{h_o}$, where $h_i$ is the height of the image and $h_o$ is the height of the object. We know $d_i=-7.5\ cm$, $d_o = 12.0\ cm$ and $h_o = 3.0\ cm$. First, find the magnification $m=-\frac{-7.5}{12}=\frac{7.5}{12}$.

Step4: Solve for $h_i$

Since $m=\frac{h_i}{h_o}$, then $h_i=m\times h_o$. Substitute $m = \frac{7.5}{12}$ and $h_o = 3.0\ cm$: $h_i=\frac{7.5}{12}\times3=\frac{22.5}{12}=1.875\ cm$.

Answer:

-7.5 1.875