A rectangular box open at the top is to be constructed so that the volume is 10 cubic meters and the base has length equal to twice its width. Material for the base of the box costs $4 persquare meter, while the material for the sides costs $2 per square meter. Our goal is to find the dimensions of the box which will minimize the cost of the box. To help you in solving this problem, follow these steps. (a) Draw a picture and label the dimensions of the box. Use h for the height, w for the width andwrite the length in terms of w. ( b ) Write an expression in terms of h and w for the cost C of the box. To do this, you need to recognize that the cost of each surface of the box is the area of that surface multiplied by thecost per square meter. Keep in mind that the box has five surfaces --- the bottom plus four sides. (c) Use the fact that the volume of the box is 10 m3 to relate h and w and then solve for h interms of w. (d) Substitute the expression for h which you found in (c) into the expression for cost in (b). (e) You should now have an expression for cost in terms of w only. Use calculus to find the dimensions of the box which will give a minimum for C analytically, find the critical value(s)of C and then use either the First or Second Derivative Test to determine where the function actually has the desired minimum. (f ) Confirm the answer which you got in part (e) by graphing the cost function on yourcalculator and estimating the value of w which minimizes the cost