The distance of satellite A from the boat X is 449 meters.
Given that:
A and B are satellites 580 meters apart.
From satellite A, there is angle of depression of 62° on boat X
From satellite B there is angle of depression of 47° on boat X
Calculations of distance between satellite A and the boat X:
Drop a perpendicular from B to AX side, and name it as K as shown in the figure given below.
Then there are two right triangles: AKB and XKB
In triangle AKB, using trigonometric ratios to find the length of KB and AK:
[tex]sin(62) = \dfrac{KB}{AB}\\\\0.883 = \dfrac{KB}{580}\\\\KB = 580 \times 0.883 = 512.11 \: \rm meters[/tex]
[tex]cos(62) = \dfrac{KA}{AB}\\\\0.47 = \dfrac{KA}{580}\\\\KA = 580 \times 0.47 = 272.6\: \rm meters[/tex]
In triangle BKX, using this value to evaluate the length KX:
Since all angles add up to 180 degrees, thus:
[tex]\angle A + \angle B + \angle X = 180\\\angle X = 180 - 62 - 47\\\angle X = 71^\circ[/tex]
And, thus we have by trigonometric ratios:
[tex]tan(71) = \dfrac{BK}{KX}\\\\2.904 = \dfrac{512.11}{KX}\\\\KX = \dfrac{512.11}{2.904}\\\\KX = 176.35 \: \rm meters[/tex]
The length of [tex]AX = AK + KX = 272.6 + 176.35 \approx 449[/tex] meters.
Thus, distance of satellite A from the boat X is 449 meters.
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