Arectangular storage container with a lid is to have a volume of 2 m3. the length of its base is twice the width. material for the base costs $1 per m2. material for the sides and lid costs $2 per m2. find the dimensions of the container which will minimize cost and the minimum cost.

Dimensions are 2 m by 1 meter by 1 meter,

Minimum cost is $ 18.

Step-by-step explanation:

Let w be the width ( in meters ) of the container,

Since, the length is twice of the width,

So, length of the container = 2w,

Now, if h be the height of the container,

Volume = length × width × height

2 = 2w × w × h

1 = w² × h

Since, the area of the base = l × w = 2w × w = 2w²,

Area of the lid = l × w = 2w²,

While the area of the sides = 2hw + 2hl

= 2h( w + l)

Since, Material for the base costs $1 per m². Material for the sides and lid costs $2 per m²,

So, the total cost,

Differentiating with respect to w,

Again differentiating with respect to w,

For maxima or minima,

C'(w) = 0

For w = 1, C''(w) = positive,

Hence, for width 1 m the cost is minimum,

Therefore, the minimum cost is C(1) = 6(1)²+12 = $ 18,

And, the dimension for which the cost is minimum is,

2 m by 1 meter by 1 meter.