1. **Problem Statement:** Find the total surface area of a square-based pyramid with vertical height and slant height in the ratio 4:5 and volume 384 cm³. 2. **Given:** - Ratio of vertical height ($h$) to slant height ($l$) is $4:5$. - Volume ($V$) = 384 cm³. 3. **Formulas:** - Volume of pyramid: $$V = \frac{1}{3} \times \text{Base Area} \times h = \frac{1}{3} s^2 h$$ where $s$ is the side length of the square base. - Slant height relation: $$l = \frac{5}{4} h$$ - Total surface area: $$A_{total} = s^2 + 2 s l$$ (base area + lateral area) 4. **Step 1: Find $h$ and $l$ in terms of $s$** From volume formula: $$384 = \frac{1}{3} s^2 h \implies h = \frac{1152}{s^2}$$ 5. **Step 2: Express $l$ in terms of $s$** $$l = \frac{5}{4} h = \frac{5}{4} \times \frac{1152}{s^2} = \frac{1440}{s^2}$$ 6. **Step 3: Use Pythagoras to relate $l$, $h$, and half base side** Since slant height $l$ is the hypotenuse of right triangle with vertical height $h$ and half base side $\frac{s}{2}$: $$l^2 = h^2 + \left(\frac{s}{2}\right)^2$$ Substitute $h$ and $l$: $$\left(\frac{1440}{s^2}\right)^2 = \left(\frac{1152}{s^2}\right)^2 + \left(\frac{s}{2}\right)^2$$ 7. **Step 4: Simplify equation:** $$\frac{2073600}{s^4} = \frac{1327104}{s^4} + \frac{s^2}{4}$$ Multiply both sides by $s^4$: $$2073600 = 1327104 + \frac{s^6}{4}$$ 8. **Step 5: Solve for $s^6$:** $$\frac{s^6}{4} = 2073600 - 1327104 = 746496$$ $$s^6 = 2985984$$ 9. **Step 6: Find $s$:** $$s = \sqrt[6]{2985984} = 8 \text{ cm}$$ 10. **Step 7: Find $h$ and $l$:** $$h = \frac{1152}{8^2} = \frac{1152}{64} = 18 \text{ cm}$$ $$l = \frac{5}{4} \times 18 = 22.5 \text{ cm}$$ 11. **Step 8: Calculate total surface area:** $$A_{total} = s^2 + 2 s l = 8^2 + 2 \times 8 \times 22.5 = 64 + 360 = 424 \text{ cm}^2$$ **Final answer:** Total surface area is $424$ cm².