1. **Problem statement:** We have two containers: one cylindrical and one spherical. Both have the same volume. The cylinder has a height of 50 cm and a radius of 11 cm. We need to find the exact radius of the spherical container. 2. **Formulas:** - Volume of a cylinder: $$V_{cyl} = \pi r^2 h$$ - Volume of a sphere: $$V_{sph} = \frac{4}{3} \pi R^3$$ 3. **Given:** - Cylinder radius $r = 11$ cm - Cylinder height $h = 50$ cm - Sphere radius $R = ?$ 4. **Step 1: Calculate the volume of the cylinder:** $$V_{cyl} = \pi \times 11^2 \times 50 = \pi \times 121 \times 50 = 6050 \pi$$ 5. **Step 2: Set the volumes equal:** Since volumes are equal, $$6050 \pi = \frac{4}{3} \pi R^3$$ 6. **Step 3: Simplify and solve for $R^3$:** Divide both sides by $\pi$: $$6050 = \frac{4}{3} R^3$$ Multiply both sides by $\frac{3}{4}$: $$R^3 = 6050 \times \frac{3}{4} = 4537.5$$ 7. **Step 4: Find $R$ by taking the cube root:** $$R = \sqrt[3]{4537.5}$$ 8. **Final answer:** The exact radius of the spherical container is $$R = \sqrt[3]{4537.5}$$ cm.