CMAS High school Geometry. Practice it free.

The three interior angles sum to $180°$: $(x+8)+(2x+17)+(x+55)=180$, so $4x+80=180$ and $x=25$.

Set up the proportion $\dfrac{x\text{ people}}{1\text{ square mile}} = \dfrac{37{,}800\text{ people}}{84\text{ square miles}}\textsf{.}$ Cross-multiplying gives $84x = 37{,}800\textsf{,}$ so $x = 450\text{ people per mile}^{2}\textsf{.}$

$\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{\dfrac{15}{17}}{\dfrac{8}{17}} = \dfrac{15}{17} \cdot \dfrac{17}{8} = \dfrac{15}{8}$.

A rotation is a rigid transformation that turns a figure around a fixed point, called a center of rotation, by a specific angle in a specific direction. All points remain equidistant from the center of rotation, keeping the image congruent to the preimage.

In a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle, each leg is the hypotenuse divided by $\sqrt{2}\textsf{:}$ $\dfrac{5\sqrt{2}}{\sqrt{2}} = 5\textsf{.}$

In a $45^{\circ}$-$45^{\circ}$-$90^{\circ}$ triangle, the hypotenuse is a leg times $\sqrt{2}\textsf{:}$ $12\sqrt{2}\textsf{.}$

Corresponding angles are congruent: $3x+16=2x+34$, so $x=18$. Each angle measures $3(18)+16=70°$.

Set up the proportion $\dfrac{12\text{ kilograms}}{1\text{ cubic meter}} = \dfrac{540\text{ kilograms}}{x\text{ cubic meters}}\textsf{.}$ Cross-multiplying gives $12x = 540\textsf{,}$ so $x = 45\text{ m}^{3}\textsf{.}$