Estimate the speed of light.
In 1675, Roemer estimated the speed of light using the following reasoning. The closest Jupiter ever gets to the earth is when the earth, the sun, and Jupitier all line up with earth in the middle. When this occurs, Jupiter is 392 million miles from the earth. The farthest Jupiter gets from the earth is when the earth, the sun, and Jupiter all line up with the sun in the middle. When this occurs, Jupiter is 575 millon miles from the earth. When Jupiter is closest to the earth, the orbits of the moons of Jupiter appear to be 16 minutes ahead of where they appear when Jupiter is farthest from the earth. Yet the length of each lunar orbit should not change according to how far Jupiter is from the earth. He realized that this apparant difference in orbit time was due to the extra time it took for the light from Jupiter's moons to reach the earth when Jupiter was farther away. Use this information to estimate the speed of light
The actual speed of light is 299,792,458 meters per second. This value is also equal to 186,282 miles per second. In physics, the speed of light is abbreviated as "c" and is one of the constants of the universe (think E=mc^2). Now let's pretend it's the 17th century and estimate the speed of light based on your word problem. Since the apparent shift in the lunar orbit is based on the time it takes the light to travel back to Earth from the two points in the Jupiter orbit, we should subtract the two distance values to get the extra distance that the light travels.
575,000,000 - 392,000,000 = 183,000,000 miles
Since the apparent lunar orbit was shifted by 16 minutes, we know that it took the light that long to travel the 183,000,000 miles. Let's convert this time into seconds.
16 minutes (60 seconds/1 minute) = 960 seconds
Finally, lets divide the two numbers to get the velocity of light.
183,000,000 miles/960 seconds = 190,625 miles per second
This is pretty close to the actual speed of light stated at the beginning of my answer (186,282 mi/s). It's only about 3% off.
speed of light in free space is 299,792,458 meter per second.