1. **State the problem:** We need to find the distance $d$ from point $X$ on the shore to point $Y$ on the island. The triangle $ZXY$ has side $ZX = 285$ meters, angle $Z = 35^\circ$, and a right angle at $Y$ between $XY$ and $ZY$. 2. **Identify the triangle type and sides:** Since $XY$ is perpendicular to $ZY$, triangle $ZXY$ is a right triangle with right angle at $Y$. 3. **Label sides relative to angle $Z$:** - Side opposite angle $Z$ is $XY = d$ (the distance we want). - Side adjacent to angle $Z$ is $ZY$. - Hypotenuse is $ZX = 285$ meters. 4. **Use the sine function:** The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ 5. **Apply the formula:** $$\sin(35^\circ) = \frac{d}{285}$$ 6. **Solve for $d$:** $$d = 285 \times \sin(35^\circ)$$ 7. **Calculate the value:** $$d = 285 \times 0.574 = 163.59$$ 8. **Round to the nearest meter:** $$d \approx 164$$ meters. **Final answer:** The distance $d$ from point $X$ to point $Y$ is approximately **164 meters**.