Question

find the side labeled x. (round your answer to one decimal place.) x =

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula for finding a side $x$ in a triangle with sides $a$, $b$ and included - angle $C$ is $x^{2}=a^{2}+b^{2}-2ab\cos C$. Here, $a = 12$, $b = 16$, and $C = 48^{\circ}$. First, find $\cos(48^{\circ})\approx0.6691$.

Step2: Substitute values into the formula

$x^{2}=12^{2}+16^{2}-2\times12\times16\times0.6691$. Calculate $12^{2}=144$, $16^{2}=256$, and $2\times12\times16\times0.6691 = 256.9344$. Then $x^{2}=144 + 256-256.9344$.

Step3: Simplify the right - hand side

$x^{2}=144+256 - 256.9344=143.0656$.

Step4: Solve for $x$

Take the square - root of both sides: $x=\sqrt{143.0656}\approx11.96$. Rounding to one decimal place, $x\approx12.0$.

Answer:

$12.0$