1. **Understanding the problem:** We have a triangle with a horizontal base labeled $z$, a left side at a $30^\circ$ angle with the base with hypotenuse length $220$ ft, a vertical segment $x$ perpendicular to the base, and the right side at a $60^\circ$ angle labeled $y$. 2. **Identify the triangle sides and angles:** - The left side forms a $30^\circ$ angle with the base, and its hypotenuse is $220$ ft. - The vertical segment $x$ is perpendicular to the base. - The right side makes a $60^\circ$ angle with the base and is labeled $y$. 3. **Calculate $x$ (the height):** Since $x$ is the height opposite the $30^\circ$ angle, $$ x = 220 \times \sin 30^\circ $$ $$ x = 220 \times \frac{1}{2} = 110 $$ ft 4. **Calculate the base segment adjacent to the $30^\circ$ angle:** This is the part of the base under the left side, $$ z_1 = 220 \times \cos 30^\circ = 220 \times \frac{\sqrt{3}}{2} = 110 \sqrt{3} $$ ft 5. **Calculate $y$ (right side length):** The right side $y$ forms a $60^\circ$ angle with the base, and its opposite side is $x = 110$ ft. Using sine, $$ y = \frac{x}{\sin 60^\circ} = \frac{110}{\frac{\sqrt{3}}{2}} = 110 \times \frac{2}{\sqrt{3}} = \frac{220}{\sqrt{3}} $$ Rationalize the denominator: $$ y = \frac{220}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{220 \sqrt{3}}{3} $$ ft 6. **Calculate the base segment under $y$ (right side base segment):** $$ z_2 = y \times \cos 60^\circ = \frac{220 \sqrt{3}}{3} \times \frac{1}{2} = \frac{110 \sqrt{3}}{3} $$ ft 7. **Calculate total base $z$:** $$ z = z_1 + z_2 = 110 \sqrt{3} + \frac{110 \sqrt{3}}{3} = 110 \sqrt{3} \left(1 + \frac{1}{3}\right) = 110 \sqrt{3} \times \frac{4}{3} = \frac{440 \sqrt{3}}{3} $$ ft **Final answers:** $$ x = 110 $$ ft $$ y = \frac{220 \sqrt{3}}{3} $$ ft $$ z = \frac{440 \sqrt{3}}{3} $$ ft