An open-top rectangular box is being constructed to hold a volume of 350 in3. The base of the box is made from a material costing 8 cents/in2. The front of the box must be decorated, and will cost 10 cents/in2. The remainder of the sides will cost 4 cents/in2. Find the dimensions that will minimize the cost of constructing this box.

the dimensions that will minimize the cost of constructing the box is:

a = 5.8481  in ;   b = 5.848 in  ; c = 10.234 in

Step-by-step explanation:

From the information given :

Let a be the base if the rectangular box

b to be the height and c to be the  other side of the rectangular box.

Then ;

the area of the base is ac

area for the front of the box is ab

area for the remaining other sides   ab + 2cb

The base of the box is made from a material costing 8 ac

The front of the box must be decorated, and will cost 10 ab

The remainder of the sides will cost 4 (ab + 2cb)

Thus ; the total cost  C is:

C = 8 ac + 10 ab + 4(ab + 2cb)

C = 8 ac + 10 ab + 4ab + 8cb

C = 8 ac + 14 ab + 8cb   ---- (1)

However; the volume of the rectangular box is V = abc = 350 in³

If abc = 350

Then b =

replacing the value for c in the above equation (1); we have :

Differentiating C with respect to a and c; we have:

--- (2)

  ---(3)

From (2)

----- (4)

From (3)

  -----(5)

Replacing the value of a in 5 into equation (4)

From (5)

  -----(5)

a = 5.8481

Recall that :

b =

b =

b =5.848

Therefore ; the dimensions that will minimize the cost of constructing the box is:

a = 5.8481  in ;   b = 5.848 in  ; c = 10.234 in