Answer
See below
Work Step by Step
Use the law of sines:
$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{y\sin C}$$
First we obtain: $\frac{b}{\sin B}=\frac{a}{\sin A}\\\sin B=\frac{\sin A}{a}\times b\\\arcsin (\sin B)=\arcsin(\frac{\sin A}{a}b)\\B=\arcsin(\frac{\sin A}{a}b)\\B=\arcsin(\frac{\sin 114^\circ}{15}\times10)\\B\approx 37^\circ$
The sum of the angles of the triangle is $180^\circ$
$$A+B+C=180^\circ\\A=180^\circ-B-C\\A=180^\circ-114^\circ-37^\circ\\A=29^\circ$$
Then $\frac{a}{\sin A}=\frac{c}{\sin C}\\c=\frac{a}{\sin A}\times\sin C=\frac{15}{\sin 114^\circ}\times\sin 29^\circ\approx7.96$
