A company packages powdered detergent in rectangular prism cartons whose bases are squares of side x cm and whose heights are h cm. The surface of the carton is coated with a special moisture barrier that costs \0.0018persquarecentimeteronthefoursidefacesand0.0026 per square centimeter on the top and bottom faces. In addition, an environmental tax of 7\% is levied on the total coating cost.
Regulations impose an upper limit of \$1.80 on the sum of coating cost and tax for each carton.
Within that financial limit, the company wishes to maximize the volume of detergent that can be shipped in a single carton. Provide complete, justified answers to the parts below.
(a) Construct an inequality that relates x and h by translating the regulatory cost ceiling into mathematical form.
(b) Solve the inequality from part (a) for h in terms of x, and identify the feasible interval for x.
(c) Express the carton volume V as a single-variable function, and compute its derivative on the open interval found in part (b).
(d) Locate all critical points of V in the feasible interval and determine the dimensions that yield the maximum volume without exceeding the cost constraint.
(e) Verify that your solution in part (d) indeed corresponds to the global maximum volume under all imposed conditions, including the cost ceiling.