Math Word Problem. Please Help.

Question

An open top box with a square base is to be constructed from 2 materials, one for the bottom and one for the sides. The volume of a box is to be 9 cubic ft. The cost of the material for the bottom is 4 pesos per sq. ft., and the cost of the material for the sides is 3 pesos per sq. ft.

 

A. Determine a model for the cost of the box as a function of its height

 

B. Find the domain of the function

Answer 1

First, read and re-read the problem until you can put it into your own words,  How's this:

   Q:  A 9-cubic-foot (fixed) box has a square base (variable) and has different costs for the base than for the sides (note: open top).

          - base costs 3 pesos/ft²

          - sides cost 4 pesos/ft²

   A.  Determine the cost as a function of its height

   B.  Determine the domain of that function

 

 

ANS:

  A volume of 9-cubic-foot is given as a constant.  That means (area of base)*height = 9 ft³.

    Or, (area of base) = (9/h) ft²       [note:  9 ft³ / h ft;   PLZ get units straight !!]

  The cost of the base is (4 pesos / ft²)(9/h ft²) = 36/h pesos

 

  Now, the (area of one side) = (side length of base)*height = ( √(9/h ft²) * (h ft) 

          = ( 3/(√h) ft ) * h ) ft

          =  3√h ft²        [note:  h/√h = √h)

  There 4 sides, each costing (3 pesos/ft²)(3√h ft²) = 4* 9√h Mex$ = 36√h pesos

 

  The total cost is (cost of base) + (cost of 4 sides):

ANS A:     Cost = 36/h + 36√h pesos

 

Checking (test case of box is 3*3*1=9 cubic feet):

    Cost should be 4*(3*3) + 3*(4*3*1) = 72 pesos   [note: 3*3 base and four 3*1 sides]

    h = 1

    Cost = 36/1 + 36√1 = 36+36 = 72 pesos

 

Checking (test case of box is 1*1*9=9 cubic feet):

   Cost should be 4*(1*1) + 3*(4*1*9) = 112 pesos

   h = 9

   Cost = 36/9 + 36√9 = 4 + 108 = 112 pesos

 

ANS B:  h≥0          [the input variable, h, is non-negative]

Answer 2

Hi Bria.

 

I wish I could draw a picture for you here...

 

The box has a square base and no top.  The material for the bottom (base) costs 4 pesos/sq ft, there is just one of these.

 

The box has four rectangular sides and the material for these sides is 3 pesos/square foot.  

 

Notice we are talking about the cost per SQUARE FOOT, this means we are talking about the cost / area.

 

Since the base is a square, all four sides will be the same length.  We will call that x.

Each side rectangle has one dimension that touches the square base, so one dimension of each rectangle will also be x.  The height of each rectangle can be any size, so we will call it h.

 

Now, you have 4 rectangles that each have dimensions (x ft  by  h ft)

You have a square base with dimensions (x feet by x feet).

 

Each rectangle costs 3 pesos per square foot, so the total cost for all four sides is (area of 1 rectangle)(4 rectangles)(3 pesos) = 12 xh.  This covers the cost of the sides.

 

The base has area (x)(x), there is only 1, and it costs 4 pesos for every square foot.  

So, the cost for the base is (x)(x)(1)(4) = Area of square base * 1 base * 4 pesos/sq ft

 

Putting these together, the cost will be 4x² + 12xh.

 

The volume of any box is the length * width * height.  Because our box has a square base, the volume equation would be x*x*h  and we are told that it is 9 cu ft.  

 

So we can write x²h = 9.

 

Now, to get the cost of the box as a function of its height, we need to replace all of the x's in the cost function with an expression with h's instead.  To do this, divide both sides of the (x²h = 9) equation by h to get that x² = h/9,

or x = √h/3   (to get x alone, take square root of both sides)

 

Now replace every x² in the (cost = 4x² + 12 xh)  with (h/9), and replace every x with √h/3

 

This results in:

 

cost = 4(h/9) + 12 (√h/3)(h)   <--- this is the answer to part A

 

The only mathematical restriction on h (the domain variable) is that h must be bigger than 0 (or equal to 0, but that would not be a box at all with no height)  so that the number under the square root radical is always ≥0.   

 

So the domain is  h≥0.  

 

So, mathematically, the height can be any positive number.  Realistically, you will probably have some cost restraint, the the maximum height would be determined by the maximum amount of money you are allowed to spend on the box.  But it doesn't seem that maximum cost information is given in the problem.  

 

Realistically,    the domain is  0 ≤ h ≤ max height as dictated by the cost.