1. **State the problem:** We are given a cone with total surface area $3584\pi$ cm², radius $r=14x$, and height $h=24x$. We need to find the volume of the cone to the nearest integer. 2. **Recall formulas:** - Curved surface area (CSA) of cone: $\pi r l$ - Volume of cone: $\frac{1}{3} \pi r^2 h$ - Total surface area (TSA) = CSA + base area = $\pi r l + \pi r^2$ 3. **Find slant height $l$ using Pythagoras:** $$l = \sqrt{r^2 + h^2} = \sqrt{(14x)^2 + (24x)^2} = \sqrt{196x^2 + 576x^2} = \sqrt{772x^2} = x\sqrt{772}$$ 4. **Write TSA in terms of $x$:** $$\text{TSA} = \pi r l + \pi r^2 = \pi (14x)(x\sqrt{772}) + \pi (14x)^2 = \pi (14x^2 \sqrt{772}) + \pi (196x^2) = \pi x^2 (14\sqrt{772} + 196)$$ 5. **Given TSA = $3584\pi$, solve for $x^2$:** $$3584\pi = \pi x^2 (14\sqrt{772} + 196)$$ Divide both sides by $\pi$: $$3584 = x^2 (14\sqrt{772} + 196)$$ $$x^2 = \frac{3584}{14\sqrt{772} + 196}$$ 6. **Calculate $14\sqrt{772} + 196$:** $$\sqrt{772} \approx 27.78$$ $$14 \times 27.78 = 389.0$$ $$389.0 + 196 = 585.0$$ 7. **Calculate $x^2$:** $$x^2 = \frac{3584}{585} \approx 6.1265$$ 8. **Calculate volume:** $$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (14x)^2 (24x) = \frac{1}{3} \pi (196 x^2)(24 x) = \frac{1}{3} \pi 4704 x^3 = 1568 \pi x^3$$ 9. **Calculate $x^3$:** $$x^3 = x \times x^2 = \sqrt{6.1265} \times 6.1265 \approx 2.476 \times 6.1265 = 15.16$$ 10. **Calculate volume numerically:** $$V = 1568 \pi \times 15.16 = 1568 \times 15.16 \pi$$ $$1568 \times 15.16 = 23774.9$$ 11. **Final volume:** $$V = 23774.9 \pi \approx 23774.9 \times 3.1416 = 74699.5$$ Rounded to nearest integer: $$\boxed{74699}$$ cm³