Question: Question 6 (4 points) a) Determine the value of $x$. $x$ $18^\circ$ $48m$ b) Determine the measure of $\angle A$ to the nearest degree. $A$ $3m$ $B$ $7m$ $C$ Show all your work! Graph shapes: Top-center: a long, thin triangle with a right angle at the upper-left vertex, $x$ marked on the left side, $18^\circ$ marked near the lower-right vertex, and $48m$ labeled along the bottom side. Center-left: a right triangle labeled $A$, $B$, $C$ with a right angle at $B$, $AB = 3m$, $AC = 7m$, and $BC$ drawn vertically down to $C$.

1. **Problem Statement:** We have two right triangles. (a) Find the length $x$ in the triangle with a right angle at the upper-left vertex, an angle of $18^\circ$ near the lower-right vertex, and the bottom side length $48m$. (b) Find the measure of $\angle A$ in the right triangle $ABC$ with right angle at $B$, where $AB=3m$ and $AC=7m$. --- 2. **Part (a): Find $x$** - The triangle is right angled at the upper-left vertex. - The side opposite the $18^\circ$ angle is $x$ (left side). - The side adjacent to the $18^\circ$ angle is $48m$ (bottom side). Using the tangent function: $$\tan(18^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{48}$$ Rearranging to solve for $x$: $$x = 48 \times \tan(18^\circ)$$ Calculate: $$x = 48 \times \tan(18^\circ) \approx 48 \times 0.3249 = 15.5952$$ Rounded to two decimal places: $$x \approx 15.60m$$ --- 3. **Part (b): Find $\angle A$** - Triangle $ABC$ is right angled at $B$. - Hypotenuse $AC = 7m$. - Side $AB = 3m$. We want to find $\angle A$. Using the cosine function (adjacent over hypotenuse): $$\cos(\angle A) = \frac{AB}{AC} = \frac{3}{7}$$ Calculate $\angle A$: $$\angle A = \cos^{-1}\left(\frac{3}{7}\right)$$ Calculate the inverse cosine: $$\angle A \approx \cos^{-1}(0.4286) \approx 64.62^\circ$$ Rounded to the nearest degree: $$\angle A \approx 65^\circ$$ --- **Final answers:** - (a) $x \approx 15.60m$ - (b) $\angle A \approx 65^\circ$