WHY ARE THEY BUILT THIS WAY? Because it's the cheapest way to do it!! We'll use calculus to show that, and we'll solve this one graphically! A propane tank is constructed by connecting (through welding, duh) two hemispheres and a cylinder together to form a tank, see above. If we add the volumes of the two hemispheres plus the cylinder we get: V=(4)/(3)pir^(3)+pir^(2)h The surface area is found similarly: SA=4pir^(2)+2pi rh [Note the "height" of the cylinder is more accurately described as its length, as it is horizontal.] The metal used to build the tank is represented as the total surface area of the tank. However, to find the material cost we must incorporate the cost of the metal for the sphere and the cylinder separately. It's hard to form metal into a spherical shape (and the metal has to be thicker), hence the material cost for the sphere is twice that of the metal used for the cylinder. Thus, the cost equation becomes: C=2**4pir^(2)+2pi rh Assuming the volume is 500 gallons, find the radius and the height (in inches!) that will minimize the cost of materials. (Don't forget to convert gallons to cubic inches!) Your answers should be reasonably close to the actual values above...note the 9^(')11^('') length above includes both radii of the hemispherical ends.