Shipping crates A square-based, box-shaped shipping crate is designed to have a volume of 16 ft . The material used to make the base costs twice as 3 much (per ft ) as the material in the sides, and the material used to make the top costs half as much (per ft") as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?

Solution Step 1: We have given the volume of a square based shipping crate; V=16ft 3 The objective is to find the dimension of the crate that minimize the cost of material the volume of a square based shipping crate; 2 V=l h V h = l h = 162 l Step 2: Let the length of the base be l. In a shipping crate designed as a square based. Let the material cost be k(is a positive constant). The material used to cost per square feet of the top is k, the cost of the sides is 2k,and the cost of the base is 4k Where the area of base is , area of the top is , and area of one side is . The cost of the material use d to make the square shaped shipping crate. cost = l 4k+l( )2k+2 k 2 l =4kl +32 +l kk 2 l =5kl +32 k l Step 3: Let the cost be y then, 2 y =5kl +32 l Now, find the dimensions of the crate that minimize the cost of materials. Find 2 Differentiate y =5kl +32 with respect to l dy d 2 k dl = dl5kl +32 ) l k =10kl+32( ) 2 l dy k So, dl=10kl+32( ) l2