Find radius and height for a bucket and what is the minimum cost?

A company wants to manufacture an open cylindrical bucket of volume 12 litres (12000cm3). The plastic used for the base of the bucket costs 0.07 cents per cm2 while the plastic used for the curved side of the bucket costs 0.05 cents per cm2. Find the radius and height of the bucket for which the bucket has minimum cost. What is the minimum cost? Show all reasoning and evaluate your answers to 2 decimal places

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Volume = pi·r^{2}·h

12,000 = pi·r^{2}·h

12,000/pi·r^{2} = h

Cost = (area of bottom)(cost of bottom material) + (area of sides)(cost of side material)

Cost = (pi·r^{2})($0.07) + (2·pi·r·h)($0.05)

Cost = 0.07·pi·r^{2} + 0.1·pi·r·h

Substitute 12,000/ pi·r^{2} in place of h (see third equation at top). This is the key step because it eliminates the variable h and reduces the cost equation to a function of a single variable, r.

Cost = 0.07·pi·r^{2} + 0.1·pi·r·(12000/pi·r^{2})

Cost = 0.07·pi·r^{2} + 1200/r

To solve, take the first derivative of Cost wrt to r, set it to zero, then solve for r. That will give you the value of r that minimizes the cost of the bucket. Plug that value of r back into the Cost equation above to find the minimum cost.

## Comments

^{2}for the material used to make the sides of the bucket? Can't set up a cost equation without it.