A company packages powdered detergent in rectangular-prism cartons with square bases…
MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A company packages powdered detergent in rectangular-prism cartons with square bases of side xcm and height hcm. Coating the four side faces costs 0.0018 dollars per square centimetre, and coating the top and bottom faces costs 0.0026 dollars per square centimetre. An environmental tax of 7% is levied on the total coating cost. Regulations impose an upper limit of 1.80 dollars on the total coating cost and tax for each carton. Within this limit, the company wishes to maximise the carton volume.
(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.