Question

A packaging company will only accept packages for shipment if the sum of its height and perimeter of the base is not more than 270 inches.

Part 1: Suppose the box you want to ship has a square base (meaning that the dimensions of the bottom of the box are equal). If you choose a box that has a length of 15 inches, how tall can the box be? (Don't forget to include units)

Part 2: What is the volume of the box you found in part a)? (Don't forget to include units)

Part 3: Write a function to represent the volume of a box with a square base, with sum of the height and perimeter of the base equal to 270 inches.

Part 4: Graph this function in an appropriate window. Show the graph below, and label completely.

Part 5: What dimensions of such a box will give the maximum volume? What is the maximum volume?

Please show the work too! Thanks

Answer 1

A packaging company will only accept backages for shipment if the sum of its height and para mater of the base is not more than 250 inches. Part-l. Box to ship has a square base of length is inches - . Then Perimeter 4X15= 60 inchos. Perimeter & height 2TO 6ot height =270 height 270-60= 210 inches Height of box 210 inches Part-2. Volume of box = area of base x height (15)2 x 210 = (47,250 cubic inches. Port-3 let side of square = x inches and height = hinchos. then uxth = 2 TO ah= 2 70-4X. Volume of box v= x.h. x² (270-4X) function V(x) = 270x² - 4x3 Part-u Graph of Vacs on graph sheet Part-s. critical point voo=0 5400-12x²=0 = 124 ( 45-2) =0 X=0 and 3 = 45 second derivative test : V"(JC) - S40-248 V"(o) 5400 Vis minimum V" (45) 540-24(45) 540-1080 - 5400 hence vis maximum. for x=45 h= 210-4445) = 270-180 =go Volume (45) x90 (182,25o Cubic inches Dimension length of square base 45 inches height of box 90 inches Maximum volume of box = 182, 250 cubic inches .