Question

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

Answer 1

Answer:

 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.  

$(AB)^2=(AC)^2+(BC)^2-2(AC)(BC)*cos (C)$

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

$x^2=1200^2+1700^2-2*1200*1700*cos (52^o)$  

$x^2=1440000+2890000-4080000*0.615661475$

$x^2=4330000-2511898.81933008$

$x^2=1818101.18066992$

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

$x=\sqrt{1818101.18066992}$

$x=1348.3698234\approx 1348$

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