Abox with a volume of 5 m3 is to be constructed with a gold-plated top, silver-plated bottom, and copper-plated sides. if gold plate costs $110 per square meter, silver plate costs $30 per square meter, and copper plate costs $14 per square meter, find the dimensions that will minimize the cost of the materials for the box.

The dimensions are 1 m × 1 m × 5 m

Explanation:

Let the dimension be x, y, z

Volume = xyz = 5 m³      ................(1)

According to question:

Cost  function, C = 110(xy) + 30(xy) + 14(2xz+2yz)

or

C = 140xy + 28xz + 28yz         ..........(2)

We need to maximise (2)

Given condition (1)

Using the concept of lagranges multipliers

   ...........(3)

             ...........(4)

        ...........(5)

From (3) and (4) and(5)

140xy + 28xz = 140xy + 28yz = 28xz + 28yz

thus,

140xy + 28xz = 140xy + 28yz

or

28xz = 28 yz

or

x = y            ............(a)

140xy + 28yz = 28xz + 28yz

substituting x from (a)

140(y)y + 28yz = 28(y)z + 28yz

or

140y² = 28yz

or

5y = z

and,

volume = 5 m³

or

xyz = 5 m³

or

x(x)(5y) = 5 m³

or

x²(5x) = 5 m³

or

5x³ = 5 m³

or

x = 1 m

Hence,

y = x = 1 m

and,

z = 5y = 5(1) = 5 m

Therefore,

The dimensions are 1 m × 1 m × 5 m

answer; i believe that the correct answer is ( owners investment)

a manufacturer of printed circuit boards uses exponential smoothing with trend to forecast monthly demand of its product. at the end of december, the company wishes to forecast sales for january. the estimate of trend through november has been 200 additional boards sold per month. average sales have been around 1000 units per month. the demand for december was 1100 units. the company uses α = 0.20 and β = 0.10. make a forecast including trend for the month of january.