Amy is building a cylindrical storage tank to hold 800 cm³ of road salt. Find the radius and height of the tank that requires the least amount of material.

Question

Amy is building a cylindrical storage tank to hold 800 cm³ of road salt. Find the radius and height of the tank that requires the least amount of material.

Answer 1

V = 800 = hpir^2

h=800/pir^2

Material needed is the surface area

A= twice the top + circumference times the height

A= 2(pi(r)^2) + hpi(2r) = 2pir^2 + 2pir(800/pir^2) = 2pir^2 +1600/r

minimize A by taking the derivative and setting equal to zero

A'(r) = 4pir -1600/r^2 = 0

multiply by r^2

4pir^3 = 1600

r^3 = 400/pi

r= the cube root of about 127.3 = a little more than 5 = radius

h= 800/25pi = about 10.2 = height