Lucy has two star systems left to visit on her voyage, but her ship is running low on fuel. the first system, ka-77, is 12001200 light years (\text{l. y.})(l. y.) away while the second system, ka-1111, is 1700\text{ l. y.}1700 l. y.. her top priority is to visit star system ka-77. to determine if she'll also be able to visit ka-1111, she must find the distance between ka-77 and ka-1111. if lucy sees an angle of 52^\circ52 ? between ka-77 and ka-1111, how far are ka-77 and ka-1111 apart? do not round during your calculations. round your final answer to the nearest light year

 1348 light years.    

Step-by-step explanation:

Please find the attachment.

Let x represent the distance between KA-7 and KA-11.

We have been given that the first system, KA-7, is 1200 light years away while the second system, KA-11, is 1700 light years away. Lucy sees an angle of 52 degrees between KA-7 and KA-11.

We can see from our attachment that Lucy, KA-7 and KA-11 forms a triangle and we will use law of cosines to solve for x.  

Upon substituting our given values in above formula we will get,

Let us take square root of both sides of our equation.

Therefore, KA-7 and KA-11 are approximately 1348 light years apart.

you didn't post any answer choices, so its not 100% clear what your teacher is looking for

however, log(15/7) is equivalent to log(15) - log(7)

i'm using the rule log(x/y) = log(x) - log(y)