Final answer:
To find the radius of a spherical tank with the same volume as a given cylindrical propane tank, equate the volume of the cylinder to the volume of the sphere and solve for the sphere's radius. The cylindrical volume formula πr²h, where r is half the diameter of 15 inches, is set equal to ⅜πr_s³, and the spherical radius r_s is isolated and solved.
Step-by-step explanation:
The equation to determine the radius of a spherical tank with the same volume as a cylindrical propane tank can be found by equating the volume formulas for both shapes and solving for the radius of the sphere. The cylinder's volume formula is V = πr²h, where r is the cylinder's radius and h is its height. The volume of a sphere is given by the formula V = ⅜πr³, where r is the sphere's radius. Given the diameter of the cylinder is 15 inches, its radius is 7.5 inches.
To find the radius r of the spherical tank, set the volume of the cylinder equal to the volume of the sphere and solve for the spherical radius:
- Find the volume of the cylinder: V_cylinder = π(7.5 inches)²(48 inches).
- Equate the two volumes and solve for the spherical radius r_s: V_cylinder = ⅜πr_s³.
- Rearrange and solve for r_s: r_s = ∛(³√(V_cylinder/⅜π)).