SOLUTION: Skyball is a popular sport in the Magical Realm where the Chaser of each team is tasked with catching the Silver Orb. In today’s match, Lina and Kai are the Chasers for opposing

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Question 1210455: Skyball is a popular sport in the Magical Realm where the Chaser of each team is tasked with catching the Silver Orb. In today’s match, Lina and Kai are the Chasers for opposing teams. Kai is floating directly above Lina when both spot the Silver Orb. In this instance, Lina’s angle of elevation to the Orb is 35°, while Kai’s angle of depression to it is 42°. Meanwhile, Orin, seated 8.2 meters directly below the Orb, watches Lina with a 28° angle of elevation. (Show your illustration)
(a) What is the distance between Lina and the Silver Orb? How about between Kai and the Orb?
(b) Orin wants Kai’s team to win, so he magically lifts the Silver Orb 1.5 meters higher. How far is Kai from the Orb now?
Answer by KMST(5330) About Me  (Show Source):
You can put this solution on YOUR website!
The drawing is not to scale because of typos, but hopefully my calculations were typo-free.
Consider right triangles KAS, LSB, and LBO. We know that SB%2B+BO=8.2m.
L%2Atan%2828%5Eo%29%2BL%2Atan%2835%5Eo%29=8.2m L%28tan%2828%5Eo%29%2Btan%2835%5Eo%29%29=8.2m L=8.2m%2F1.231917=6.656293m
The distance between Lina and the Orb is
L%2Fcos%2835%5Eo%29=6.656293m%2F0.819152=highlight%288.126m%29
The distance between Kai and the Orb is
L%2Fcos%2842%5Eo%29=6.656293m%2F0.743145=highlight%288.957m%29
If Orin magically lifts the Silver Orb 1.5 meters higher, it will be at point %22S%27%22, 1.5 meters above S.
In triangle KAS, AS=L%2Atan%2842%5Eo%29=6.656293m%2A0.900404=5.993353m.
In the new right triangle %22KAS%27%22, %22AS%27%22=5.993353m-1.5m=4.493353m.
The length of side KA is still 6.656293m.
Kai's distance to %22S%27%22 in meters can be calculated from the Pythagorean theorem as
%22KS%27%22=sqrt%286.65629%5E2%2B4.493353%5E2%29=8.030968413 highlight%288.031m%29 when rounded to 3 decimal places. That puts Kai closer to the Silver Orb than Lina.
Orin could instantly calculate all that. His magical powers make him a geometry and trigonometry wizard too.