Michele wanted to measure the height of her school's flagpole. She placed a mirror on the ground 60 ft from the flag pole, then walked backward until she was able to see the top of the pole in the mirror. Her eyes are 5 feet above the ground and she was 12 feet from the mirror. Using similar triangles, find the height of the flag pole to the nearest tenth of a foot.
Let's represent the height of the flagpole as $h$.
We can form similar triangles between Michele, the mirror, and the top of the flagpole.
[asy]
pair A,B,C;
A = (-60,0);
B = (0,0);
C = (0, 19);
draw(A--B--C--cycle);
draw(B--(-5,0)--C); label("60 ft",B--A,S);
label("5 ft",(0,0)--(0,5),W);
label("12 ft",(0,0)--(12,0),S);
[/asy]
We can set up a proportion to find the height of the flagpole:
$\frac{h-5}{12}=\frac{h}{60}.$
Cross-multiplying gives $60(h-5)=12h.$
Expanding gives $60h-300=12h.$
Subtracting $12h$ from both sides gives $48h-300=0.$
Adding $300$ to both sides gives $48h=300.$
Dividing both sides by $48$ gives $h=\frac{300}{48}=\frac{25}{4}=6.25.$
Therefore, the height of the flagpole is $\boxed{6.25}$ feet.