A manufacturer forms an open-top box by cutting four equal squares from the corners…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A manufacturer forms an open-top box by cutting four equal squares from the corners of a rectangular sheet of cardboard that measures 24 inches by 36 inches, then folding up the sides. Let the side length of each removed square be x inches.
(a) Express the volume V of the box in terms of x.
(b) State the interval of x values that is physically meaningful for this construction and explain your reasoning.
(c) Find every critical value of V on the interval from part (b).
(d) Determine the value of x that produces the greatest possible volume and calculate that maximum volume to the nearest cubic inch.
(e) Verify that the volume obtained in part (d) is indeed a maximum within the domain, addressing both the critical points and any relevant endpoints.