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 $x = 450\text{ people per mile}^{2}\textsf{.}$

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

Both legs of a $45$-$45$-$90$ triangle are equal, so the other leg is $15$.

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 $x = 45\text{ m}^{3}\textsf{.}$

$\cos\alpha = \dfrac{\sin\alpha}{\tan\alpha} = \dfrac{\dfrac{4}{5}}{\dfrac{4}{3}} = \dfrac{4}{5} \cdot \dfrac{3}{4} = \dfrac{3}{5}$.

The hypotenuse is twice the short leg: $2 \times 10 = 20$.