Answer:
[tex]b =\boxed{\quad 10.6 \quad}[/tex]
Step-by-step explanation:
The law of sines already provided in the question states that the ratio of the side of a triangle to the sine of the angle opposite that side is the same for all sides
Mathematically,
[tex]\dfrac{a}{\sin A} = \dfrac{b}{\sin C} = \dfrac{c}{\sin C}[/tex]
where
a, b, c are the three sides and
A, B, C are the angles opposite each of the sides respectively
We are given one of the sides as 15 and the angle opposite as 40°
The other side is represented by the letter b and the angle opposite is 27°
Applying the law of sines to these two sides and their opposite angles we get
[tex]\dfrac{b}{\sin 27} = \dfrac{15}{\sin 40}[/tex]
Multiplying both sides by sin 27 gives
[tex]\dfrac{b}{\sin 27} \cdot \sin 27 = \dfrac{15}{\sin 40} \cdot \sin 27\\\\b = \dfrac{15}{\sin 40} \cdot \sin 27\\\\[/tex]
Using a calculator this works out to
[tex]b = 10.59425\dots[/tex]
Rounded to nearest tenth
[tex]b = 10.6[/tex]