1. **State the problem:** We need to find the distance across the lake, which is the length of segment $QR$. 2. **Given:** - $PS = 180$ m - $SQ = 70$ m - $ST = 147.6$ m 3. **Understand the setup:** Points $P$, $S$, and $T$ form a triangle with right angles at $Q$ and $S$. We have two right triangles: $PQS$ and $PST$. 4. **Goal:** Find $QR$, which is the vertical distance across the lake. 5. **Use the Pythagorean theorem:** - In triangle $PQS$, right angled at $Q$, we have: $$PQ^2 = PS^2 + SQ^2$$ $$PQ^2 = 180^2 + 70^2 = 32400 + 4900 = 37300$$ $$PQ = \sqrt{37300} \approx 193.15 \text{ m}$$ 6. **In triangle $PST$, right angled at $S$, we have:** $$PT^2 = PS^2 + ST^2$$ $$PT^2 = 180^2 + 147.6^2 = 32400 + 21777.76 = 54177.76$$ $$PT = \sqrt{54177.76} \approx 232.79 \text{ m}$$ 7. **Find $QR$:** Since $QR$ is the vertical distance across the lake, and $PQ$ and $PT$ are distances from $P$ to $Q$ and $T$ respectively, the difference in vertical components corresponds to $QR$. 8. **Calculate $QR$ as:** $$QR = PT - PQ = 232.79 - 193.15 = 39.64 \text{ m}$$ 9. **Final answer:** $$\boxed{39.64 \text{ meters}}$$ This is the distance across the lake, rounded to the nearest hundredth of a meter.