find the measure of the acute angle, x, in degrees. the figure is drawn to scale. round to the hundredths…

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula for finding an angle in a triangle is $\cos(x)=\frac{b^{2}+c^{2}-a^{2}}{2bc}$. Let $a = 6$, $b = 10$, and the included - angle of $b$ and $c$ is $36^{\circ}$. First, find the third - side using the Law of Cosines: $a^{2}=b^{2}+c^{2}-2bc\cos(A)$. But we want to find $\cos(x)$, so $\cos(x)=\frac{10^{2}+6^{2}-c^{2}}{2\times10\times6}$. First, find $c$ using the Law of Cosines with the given angle $A = 36^{\circ}$, $b = 10$, and $a = 6$: $c^{2}=10^{2}+6^{2}-2\times10\times6\times\cos(36^{\circ})$. $c^{2}=100 + 36-120\times\cos(36^{\circ})$. $\cos(36^{\circ})\approx0.8090$, so $c^{2}=136-120\times0.8090=136 - 97.08 = 38.92$, and $c=\sqrt{38.92}\approx6.24$. Now, use the Law of Cosines to find $\cos(x)$: $\cos(x)=\frac{10^{2}+6.24^{2}-6^{2}}{2\times10\times6.24}=\frac{100 + 38.94 - 36}{124.8}=\frac{102.94}{124.8}\approx0.825$.

Step2: Find the angle $x$

Since $\cos(x)\approx0.825$, then $x=\cos^{-1}(0.825)$. $x=\cos^{-1}(0.825)\approx34.44^{\circ}$.

Answer:

$34.44^{\circ}$