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Problem with the triangle PQT (altitude problem)

Part A

Answer is 60 degrees. 

You can solve x+30+90 = 180 or you can solve x+30 = 90 to find that x = 60. Note how angle PTR and angle TPR are complementary angles. A shortcut is to do 90-30 = 60.



Part B

The exact length of PR is 7 inches 

Triangle TPR is a 30-60-90 triangle, so the short leg is half that of the hypotenuse (14). Take half of 14 to get 7. The short leg is always opposite the 30 degree angle for a 30-60-90 triangle.



Part C

7*sqrt(3)
The notation "sqrt" is shorthand for "square root"

TR is the long leg of the right triangle TPR. The long leg of a 30-60-90 triangle is equal to sqrt(3) times the short leg. Alternatively, you can use the pythagorean theorem to solve for the long leg.



Part D

7*sqrt(2)

Triangle PRQ is a right triangle. It is a 45-45-90 right triangle. The hypotenuse is equal to sqrt(2) times the leg 7. Alternatively, you can use the pythagorean theorem to solve for the hypotenuse.



Problem with the tree and support wire

Part A

36.9

Work Shown:
cos(x) = adjacent/hypotenuse
cos(x) = 24/30
x = arccos(24/30)
x = 36.8698976
x = 36.9



Part B

18

Use the pythagorean theorem to find that
a^2 + b^2 = c^2
a^2 + 24^2 = 30^2
a^2 + 576 = 900
a^2 = 900-576
a^2 = 324
a = sqrt(324)
a = 18