Question

the 200 - inch - diameter concave mirror of the hale telescope on mount palomar has a focal length of 16.9 m. an astronomer stands 20.0 m in front of this mirror. what is the magnification of her image?

a. - 1.2
b. - 3.9
c. 5.5
d. - 8.3
e. - 9.7

Explanation:

Step1: Recall 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. Given $f = 16.9\ m$ and $d_o=20.0\ m$.

Step2: Solve for $d_i$

First, rewrite the mirror formula for $d_i$: $\frac{1}{d_i}=\frac{1}{f}-\frac{1}{d_o}$. Substitute $f = 16.9\ m$ and $d_o = 20.0\ m$ into the formula. $\frac{1}{d_i}=\frac{1}{16.9}-\frac{1}{20.0}=\frac{20.0 - 16.9}{16.9\times20.0}=\frac{3.1}{16.9\times20.0}$. Then $d_i=\frac{16.9\times20.0}{3.1}\approx109.03\ m$.

Step3: Recall magnification formula

The magnification formula is $m=-\frac{d_i}{d_o}$.

Step4: Calculate magnification

Substitute $d_i\approx109.03\ m$ and $d_o = 20.0\ m$ into the magnification formula: $m=-\frac{109.03}{20.0}\approx - 5.5$.

Answer:

C. - 5.5