1. **Problem 1:** Find the radius $r$ of a cone with height $h=12$ km and volume $V=100\pi$ km³. The volume formula for a cone is: $$V=\frac{1}{3}\pi r^2 h$$ Substitute known values: $$100\pi=\frac{1}{3}\pi r^2 \times 12$$ Divide both sides by $\pi$: $$100=\frac{1}{3} r^2 \times 12$$ Simplify: $$100=4 r^2$$ Divide both sides by 4: $$\cancel{4} r^2=\frac{100}{\cancel{4}}$$ $$r^2=25$$ Take the square root: $$r=\sqrt{25}=5$$ Radius $r=5$ km. 2. **Problem 2:** Find the diameter $d$ of a cone with height $h=9$ cm and volume $V=462$ cm³. Volume formula: $$V=\frac{1}{3}\pi r^2 h$$ Substitute values: $$462=\frac{1}{3}\pi r^2 \times 9$$ Simplify: $$462=3\pi r^2$$ Divide both sides by $3\pi$: $$\frac{462}{3\pi}=r^2$$ Calculate: $$r^2=\frac{462}{3\pi}=\frac{462}{9.4248}\approx49$$ Take square root: $$r=\sqrt{49}=7$$ Diameter $d=2r=14$ cm. 3. **Problem 3:** Find the radius $r$ of a cylinder with height $h=12$ m and volume $V=2400$ m³. Volume formula for cylinder: $$V=\pi r^2 h$$ Substitute values: $$2400=\pi r^2 \times 12$$ Divide both sides by $12\pi$: $$r^2=\frac{2400}{12\pi}$$ Simplify: $$r^2=\frac{2400}{37.6991}\approx63.66$$ Take square root: $$r=\sqrt{63.66}\approx8$$ Radius $r=8$ m. 4. **Problem 4:** Find the diameter $d$ of a cylinder with height $h=4$ ft and volume $V=484\pi$ ft³. Volume formula: $$V=\pi r^2 h$$ Substitute values: $$484\pi=\pi r^2 \times 4$$ Divide both sides by $\pi$: $$484=r^2 \times 4$$ Divide both sides by 4: $$\cancel{4} r^2=\frac{484}{\cancel{4}}$$ $$r^2=121$$ Take square root: $$r=\sqrt{121}=11$$ Diameter $d=2r=22$ ft. 5. **Problem 5:** Find the radius $r$ of a sphere with volume $V=3050$ in³. Volume formula for sphere: $$V=\frac{4}{3}\pi r^3$$ Substitute values: $$3050=\frac{4}{3}\pi r^3$$ Divide both sides by $\frac{4}{3}\pi$: $$r^3=\frac{3050}{\frac{4}{3}\pi}=\frac{3050 \times 3}{4\pi}$$ Calculate denominator: $$4\pi=12.5664$$ Calculate numerator: $$3050 \times 3=9150$$ Divide: $$r^3=\frac{9150}{12.5664}\approx728.1$$ Take cube root: $$r=\sqrt[3]{728.1}\approx9$$ Radius $r=9$ in. 6. **Problem 6:** Find the diameter $d$ of a sphere with volume $V=2304\pi$ units³. Volume formula: $$V=\frac{4}{3}\pi r^3$$ Substitute values: $$2304\pi=\frac{4}{3}\pi r^3$$ Divide both sides by $\pi$: $$2304=\frac{4}{3} r^3$$ Multiply both sides by $\frac{3}{4}$: $$\cancel{\frac{4}{3}} r^3=2304 \times \frac{3}{\cancel{4}}$$ $$r^3=2304 \times \frac{3}{4}=1728$$ Take cube root: $$r=\sqrt[3]{1728}=12$$ Diameter $d=2r=24$ units. **Sum of all answers:** $$5 + 14 + 8 + 22 + 9 + 24 = 82$$ **Final answer:** 82