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practical trigonometry

heights and distance
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practical trigonometry heights and distance


A vertical tree is partially broken by the wind . The top of
the tree still connected to the lower portion
 touches the ground and makes an angle
30 degrees  with the ground. If the top of the tree touches
the ground 30 m away from the foot of the tree, what
was the height of the tree before it broke

heights
CAB = full tree broken at A
C = top of the tree now touching the ground
B = foot of the tree

tan(30 degrees) = AB/CB
1/[sqrt(3)] = AB/30
AB= 30 / sqrt(3) introduce conjugate
AB = 30*sqrt(3) / 3 = 10*sqrt(3) m --------(1)

cos(30 degrees)= BC/AC
[sqrt(3)]/2 = 30 / AC
AC = 30*2/[sqrt(3)] introduce conjugate
AC= 60sqrt(3)/3 = 20*sqrt(3) m ----------(2)

Total height of the tree before it broke
= AB+AC = 10*sqrt(3)+20*sqrt(3)
=30*sqrt(3) m


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problem 2
  

Find the sum of all 3 digit natural numbers
which are divisible by 8

The smallest 3 digit number is 100
dividing 100 with 8, quotient is 12 with
remainder 4
therefore 13*8 =104 is the first term
Similarly dividing 999 with 8 quotient is 124
with remainder 7
124*8 = 992 is the last term

series is 104 + 112 + ...+992
a = 104
l=992
d=8 (common difference)
number of terms n = [(l-a)/d]+1
n = [(992-104)/8]+1=112

sum = [n/2][a+l]
=[112/2]*[104+992]=61376




other questions and problems:


*integral calculus integration formula





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trigonometric identity and ratio of  certain standard angles

*list of differentiation formula     *integral calculus integration formula
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