Question 1195042
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A vertical pole on a 15° slope is braced by a wire from the top of the pole 
to a point 20 ft. uphill from the base. If the pole subtends an angle of 62°30' 
from this point, find the height of the pole and length of the wire.
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<pre>
Let point A be the base of the pole;
point B be the upper end of the pole,
and point C be the point where the wire is attached to the slope of the hill.


We have triangle ABC with this given information:

    - angle C is 62°30';

    - angle A is 90° - 15° = 75°;

    - side AC is 20 ft long.


They want you find side c = AB (the pole height) and the side a = BC (the length of the wire).


First, we find the measure of the angle B: it is 180° - 62°30' - 75° = 42°30'.


Next, write sine law proportion

    {{{a/sin(A)}}} = {{{b/sin(B)}}} = {{{c/sin(C)}}}.



To find side c = AB, use this part of the proportion

    {{{b/sin(B)}}} = {{{c/sin(C)}}},  {{{b/sin(42.5^o)}}} = {{{c/sin(62.5^o)}}},  {{{20/0.67559}}} = {{{c/0.88701}}},

    which gives  c = {{{(20*0.88701)/0.67559}}} = 26.26 ft  (the height of the pole).     <U>ANSWER</U>



To find side a = BC, use this part of the proportion

    {{{b/sin(B)}}} = {{{a/sin(A)}}},  {{{b/sin(42.5^o)}}} = {{{a/sin(75^o)}}},  {{{20/0.67559}}} = {{{a/0.9659}}},

    which gives  a = {{{(20*0.9659)/0.67559}}} = 28.59 ft  (the length of the wire).      <U>ANSWER</U>
</pre>

Solved.