A rectangular wall cabinet with a mirror in the center and six movable shelves (three on each side) is to be constructed using 5" wide wood planks as shown in Figure 2a above. The cabinet is to be designed so that the area of the mirror (with base b/2 and height h) will be exactly 400 square inches.
The following materials are used:
Four outside boards (A,B,C,D) plus two vertical dividers (E,F), each purchased at a cost of $0.25 per linear inch.
Six shelf boards (G,H,I,J,K,L), each one-fourth the width of the cabinet and purchased at a cost of $0.22 per linear inch.
Eight metal shelf tracks to be installed on the insides of boards A,C,E and F facing the horizontal shelves, two tracks per board. (See Figure 2b. The six shelves will be attached to these tracks.) Each single track must be exactly 3/4 the height of the cabinet and will be purchased at a cost of $0.08 per linear inch.
A rectangular glass mirror, cut to the full height of the cabinet but only one-half of the width, purchased at a cost of $0.06 per square inch.
Find the overall dimensions that will minimize the total cost of the materials. (You may neglect the thickness of the boards.) Express your optimal dimensions in inches (accurate to three decimal places). Also determine the total cost of the materials for this design, rounded to the nearest cent.
The following materials are used:
Four outside boards (A,B,C,D) plus two vertical dividers (E,F), each purchased at a cost of $0.25 per linear inch.
Six shelf boards (G,H,I,J,K,L), each one-fourth the width of the cabinet and purchased at a cost of $0.22 per linear inch.
Eight metal shelf tracks to be installed on the insides of boards A,C,E and F facing the horizontal shelves, two tracks per board. (See Figure 2b. The six shelves will be attached to these tracks.) Each single track must be exactly 3/4 the height of the cabinet and will be purchased at a cost of $0.08 per linear inch.
A rectangular glass mirror, cut to the full height of the cabinet but only one-half of the width, purchased at a cost of $0.06 per square inch.
Find the overall dimensions that will minimize the total cost of the materials. (You may neglect the thickness of the boards.) Express your optimal dimensions in inches (accurate to three decimal places). Also determine the total cost of the materials for this design, rounded to the nearest cent.