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Answer: 42.21 km
Step-by-step explanation:
We can solve this using trigonometry, since we have the following data:
is the the angle of elevation
is the horizontal distance between the plane and the radar station
is the hypotenuse of the right triangle formed between the radar station and the airplane
Now, the trigonometric function that will be used is <u>cosine</u>:
because
is the adjacent side of the right triangle
Finding :
Check the picture below.
so the perimeter of the polygon is the sum of all its sides, namely, AB + BC + CD + DA.
now, let's check how long each side is,
![\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\ \quad \\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &A&(~{{ -6}} &,&{{ -4}}~) % (c,d) &B&(~{{ -3}} &,&{{ 6}}~) \end{array} \\\\\\ d = \sqrt{({{ x_2}}-{{ x_1}})^2 + ({{ y_2}}-{{ y_1}})^2}\\\\ -------------------------------\\\\ AB=\sqrt{[-3-(-6)]^2+[6-(-4)]^2} \\\\\\ AB=\sqrt{(-3+6)^2+(6+4)^2} \\\\\\ AB=\sqrt{3^2+10^2}\implies \boxed{AB=\sqrt{109}}\\\\ -------------------------------](https://tex.z-dn.net/?f=%5Cbf%20~~~~~~~~~~~~%5Ctextit%7Bdistance%20between%202%20points%7D%5C%5C%20%5Cquad%20%5C%5C%0A%5Cbegin%7Barray%7D%7Bccccccccc%7D%0A%26%26x_1%26%26y_1%26%26x_2%26%26y_2%5C%5C%0A%25%20%20%28a%2Cb%29%0A%26A%26%28~%7B%7B%20-6%7D%7D%20%26%2C%26%7B%7B%20-4%7D%7D~%29%20%0A%25%20%20%28c%2Cd%29%0A%26B%26%28~%7B%7B%20-3%7D%7D%20%26%2C%26%7B%7B%206%7D%7D~%29%0A%5Cend%7Barray%7D%0A%5C%5C%5C%5C%5C%5C%0Ad%20%3D%20%5Csqrt%7B%28%7B%7B%20x_2%7D%7D-%7B%7B%20x_1%7D%7D%29%5E2%20%2B%20%28%7B%7B%20y_2%7D%7D-%7B%7B%20y_1%7D%7D%29%5E2%7D%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0AAB%3D%5Csqrt%7B%5B-3-%28-6%29%5D%5E2%2B%5B6-%28-4%29%5D%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0AAB%3D%5Csqrt%7B%28-3%2B6%29%5E2%2B%286%2B4%29%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0AAB%3D%5Csqrt%7B3%5E2%2B10%5E2%7D%5Cimplies%20%5Cboxed%7BAB%3D%5Csqrt%7B109%7D%7D%5C%5C%5C%5C%0A-------------------------------)
![\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\ \quad \\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &B&(~{{ -3}} &,&{{6}}~) % (c,d) &C&(~{{ 4}} &,&{{ 0}}~) \end{array} \\\\ -------------------------------\\\\ BC=\sqrt{[4-(-3)]^2+[0-6]^2}\implies BC=\sqrt{(4+3)^2+(0-6)^2} \\\\\\ BC=\sqrt{7^2+(-6)^2}\implies \boxed{BC=\sqrt{85}}\\\\ -------------------------------](https://tex.z-dn.net/?f=%5Cbf%20~~~~~~~~~~~~%5Ctextit%7Bdistance%20between%202%20points%7D%5C%5C%20%5Cquad%20%5C%5C%0A%5Cbegin%7Barray%7D%7Bccccccccc%7D%0A%26%26x_1%26%26y_1%26%26x_2%26%26y_2%5C%5C%0A%25%20%20%28a%2Cb%29%0A%26B%26%28~%7B%7B%20-3%7D%7D%20%26%2C%26%7B%7B6%7D%7D~%29%20%0A%25%20%20%28c%2Cd%29%0A%26C%26%28~%7B%7B%204%7D%7D%20%26%2C%26%7B%7B%200%7D%7D~%29%0A%5Cend%7Barray%7D%0A%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0ABC%3D%5Csqrt%7B%5B4-%28-3%29%5D%5E2%2B%5B0-6%5D%5E2%7D%5Cimplies%20BC%3D%5Csqrt%7B%284%2B3%29%5E2%2B%280-6%29%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0ABC%3D%5Csqrt%7B7%5E2%2B%28-6%29%5E2%7D%5Cimplies%20%5Cboxed%7BBC%3D%5Csqrt%7B85%7D%7D%5C%5C%5C%5C%0A-------------------------------)
![\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\ \quad \\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &D(~{{ 2}} &,&{{-1}}~) % (c,d) &A&(~{{ -6}} &,&{{ -4}}~) \end{array}\\\\ -------------------------------\\\\ DA=\sqrt{[-6-2]^2+[-4-(-1)]^2}\\\\\\ DA=\sqrt{(-6-2)^2+(-4+1)^2} \\\\\\ DA=\sqrt{(-8)^2+(-3)^2}\implies \boxed{DA=\sqrt{73}}](https://tex.z-dn.net/?f=%5Cbf%20~~~~~~~~~~~~%5Ctextit%7Bdistance%20between%202%20points%7D%5C%5C%20%5Cquad%20%5C%5C%0A%5Cbegin%7Barray%7D%7Bccccccccc%7D%0A%26%26x_1%26%26y_1%26%26x_2%26%26y_2%5C%5C%0A%25%20%20%28a%2Cb%29%0A%26D%28~%7B%7B%202%7D%7D%20%26%2C%26%7B%7B-1%7D%7D~%29%20%0A%25%20%20%28c%2Cd%29%0A%26A%26%28~%7B%7B%20-6%7D%7D%20%26%2C%26%7B%7B%20-4%7D%7D~%29%0A%5Cend%7Barray%7D%5C%5C%5C%5C%0A-------------------------------%5C%5C%5C%5C%0ADA%3D%5Csqrt%7B%5B-6-2%5D%5E2%2B%5B-4-%28-1%29%5D%5E2%7D%5C%5C%5C%5C%5C%5C%20DA%3D%5Csqrt%7B%28-6-2%29%5E2%2B%28-4%2B1%29%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0ADA%3D%5Csqrt%7B%28-8%29%5E2%2B%28-3%29%5E2%7D%5Cimplies%20%5Cboxed%7BDA%3D%5Csqrt%7B73%7D%7D)
sum those sides up, and that's the perimeter of the polygon.
so the perimeter of the polygon is the sum of all its sides, namely, AB + BC + CD + DA.
now, let's check how long each side is,
sum those sides up, and that's the perimeter of the polygon.