1. **Problem statement:** A canvas wind shelter has a back, two square sides, and a top. The total canvas area is $147 \frac{1}{2}$ square feet. We want to find the depth that maximizes the volume inside the shelter. 2. **Define variables and formulas:** Let the depth be $d$ feet. Since the sides are square, let the side length be $s$ feet. The canvas area consists of: - Back: $s \times d = sd$ - Two square sides: each $s \times s = s^2$, so total $2s^2$ - Top: $s \times d = sd$ Total canvas area $A$ is: $$A = sd + 2s^2 + sd = 2sd + 2s^2$$ Given $A = 147.5$ (which is $147 \frac{1}{2}$), so: $$2sd + 2s^2 = 147.5$$ Divide both sides by 2: $$sd + s^2 = 73.75$$ 3. **Express $d$ in terms of $s$:** $$d = \frac{73.75 - s^2}{s} = \frac{73.75}{s} - s$$ 4. **Volume $V$ inside the shelter:** $$V = \text{area of base} \times \text{depth} = s^2 \times d = s^2 \left(\frac{73.75}{s} - s\right) = 73.75 s - s^3$$ 5. **Maximize volume $V$:** Take derivative with respect to $s$: $$\frac{dV}{ds} = 73.75 - 3s^2$$ Set derivative to zero for critical points: $$73.75 - 3s^2 = 0 \implies 3s^2 = 73.75 \implies s^2 = \frac{73.75}{3} \approx 24.5833$$ $$s = \sqrt{24.5833} \approx 4.958$$ 6. **Find corresponding depth $d$:** $$d = \frac{73.75}{4.958} - 4.958 \approx 14.88 - 4.958 = 9.922$$ 7. **Check if this matches any answer choices:** No answer choice matches $d \approx 9.922$ feet. 8. **Re-examine problem:** Sides are square, so $s$ is side length of square sides, depth is $d$. Canvas area is $2sd + 2s^2 = 147.5$. Volume is $s^2 d$. Try to express volume in terms of $d$ instead: From $2sd + 2s^2 = 147.5$: $$sd + s^2 = 73.75$$ Rewrite as: $$s^2 + s d - 73.75 = 0$$ Solve quadratic in $s$: $$s = \frac{-d \pm \sqrt{d^2 + 4 \times 73.75}}{2}$$ Since $s$ is positive, take positive root: $$s = \frac{-d + \sqrt{d^2 + 295}}{2}$$ Volume: $$V = s^2 d = d \left(\frac{-d + \sqrt{d^2 + 295}}{2}\right)^2$$ 9. **Maximize $V(d)$:** This is complicated, but we can test answer choices for $d$: - (A) $d=\frac{7}{2} = 3.5$ - (B) $d=\frac{7}{4} = 1.75$ - (C) $d=4$ - (D) $d=7$ Calculate $V$ for each: For $d=3.5$: $$s = \frac{-3.5 + \sqrt{3.5^2 + 295}}{2} = \frac{-3.5 + \sqrt{12.25 + 295}}{2} = \frac{-3.5 + \sqrt{307.25}}{2} \approx \frac{-3.5 + 17.53}{2} = 7.015$$ $$V = s^2 d = 7.015^2 \times 3.5 = 49.21 \times 3.5 = 172.24$$ For $d=1.75$: $$s = \frac{-1.75 + \sqrt{1.75^2 + 295}}{2} = \frac{-1.75 + \sqrt{3.06 + 295}}{2} = \frac{-1.75 + 17.18}{2} = 7.715$$ $$V = 7.715^2 \times 1.75 = 59.54 \times 1.75 = 104.15$$ For $d=4$: $$s = \frac{-4 + \sqrt{16 + 295}}{2} = \frac{-4 + 17.32}{2} = 6.66$$ $$V = 6.66^2 \times 4 = 44.36 \times 4 = 177.44$$ For $d=7$: $$s = \frac{-7 + \sqrt{49 + 295}}{2} = \frac{-7 + 18.03}{2} = 5.515$$ $$V = 5.515^2 \times 7 = 30.42 \times 7 = 212.94$$ 10. **Conclusion:** Maximum volume occurs at $d=7$ feet, which is choice (D). --- **Final answer:** $$\boxed{7 \text{ feet}}$$