Answer:
the answer is D
Explanation:
According to the Law of Cosines, for any △ABC
with side lengths a
, b
, and c
, a2=b2+c2−2bccosA
; b2=a2+c2−2accosB
; and c2=a2+b2−2abcosC
The figure shows the same towers and boat as in the beginning of the task. The sides of the triangle, the lengths of the sides and angle A are highlighted in red.
Set up the equation for the Law of Cosines. Substitute x
for the length of the unknown side and substitute the known values into the Law of Cosines:
x2=30002+15382−2(3000)(1538)cos35°
Square the values and multiply:
x2=9000000+2365444−9228000cos35°
Add:
x2=11365444−9228000cos35°
Take the square root of both sides:
x=11365444−9228000cos35°−−−−−−−−−−−−−−−−−−−−−−√
Calculate the value of cosine 35°
on a calculator and solve for the positive square root to find x
x≈1951 yd
Therefore, x≈1951 yd
So, the distance from the boat to the observer in tower B
is about 1951 yd
Step 2
According to the Law of Sines, for any △ABC
with side lengths a
, b
, and c
, sinAa=sinBb=sinCc
The figure shows the same towers and boat as in the beginning of the task. The distance between point B and the boat is 1951 yards. The sides of the triangle, except side A B, and angles A and B are highlighted in red.
By the Law of Sines, set up the proportion sin A
is to the distance from the boat to the observer in tower B
as sin B
is to the distance from the boat to the observer in tower A
. Substitute the known values:
sin35°1951=sinB1538
Cross multiply:
1951(sinB)=1538(sin35°)
Divide both sides by 1951
sinB=15381951sin35°
According to this equation m∠B
is equal to the inverse sine function of 15381951sin35°
. Write this:
m∠B=sin−1(15381951sin35°)
Calculate the value of sine 35°
on a calculator, substitute and calculate the value of the inverse sine 15381951sin35°
on a calculator:
m∠B≈27°
Therefore, m∠B≈27°
Hope this long explanation helps, lol!