the volume of a cylindrical can is 500cm^3. The materail used to make the top and bottom costs 0.012 cent/cm^2 the material used for the sides costs 0.01 cent?cm^2, and the seam joining the top and bottom to the sides costs .015 cent/cm. what size can would cost the least to produce?

Answer:radius: 3.671 cmheight: 11.810 cmStep-by-step explanation:The total cost of producing a cylindrical can with radius r and height h will be ...   cost = (lateral area)×(side cost) +(end area)×(end cost) +(seam length)×(seam cost)__The lateral area (LA) is ...   LA = 2πrhSince the volume of the can is fixed, we can write the height in terms of the radius using the volume formula.   V = πr²h   h = V/(πr²)Then the lateral area is ...   LA = 2πr(V/(πr²)) = 2V/r = 2·500/r = 1000/r__The end area (EA) is twice the area of a circle of radius r:   EA = 2×(πr²) = 2πr²__The seam length (SL) is twice the circumference of the end:   SL = 2×(2πr) = 4πr__So, the total cost in cents of producing the can, in terms of its radius, is ...   cost = (1000/r)(0.01) +(2πr²)(.012) +(4πr)(0.015)We can find the minimum by setting the derivative to zero.   d(cost)/dr = -10/r² +0.048πr +.06π = 0Multiplying by r² gives the cubic ...   0.048πr³ +0.06πr² -10 = 0   r³ +1.25r² -(625/(3π)) = 0 . . . . . . divide by .048πThis can be solved graphically, or using a spreadsheet to find the value of r to be about 3.671 cm. The corresponding value of h is ...   h = 500/(π·3.671²) ≈ 11.810 . . . cmThe minimum-cost can will have a radius of about 3.671 cm and a height of about 11.810 cm._____A graphing calculator can find the minimum of the cost function without having to take derivatives and solve a cubic.