Please, give an appropriate mathematical statement for the following problem

Question

An open box is to be formed out of a rectangle piece of cardboard whose length is 12cm longer than its width. To form the box, a square of side 5 cm will be removed from each corner of the cardboard. Then the edges of the remaining cardboard will be turned up. If the box was to hold at most 1900 cm^3, what mathematical statement would represent the given situation?

Answer 1

length:  x cm

width:  x - 12 cm

height:  5 cm

 

Volume = L*W*H = 5x(x-12) = (5x² - 60x) cm³

 

The box can hold "AT MOST" 1900 cm meaning it can hold from 0 to 1900 cm³ giving us:

 

5x² - 60x ≤ 1900

Factoring out a 5 this can be written as:  x² - 12x ≤ 380

 

Working this as a quadratic to find the maximum dimensions:

 

x² - 12x - 380 = 0

 

x = [12 ±√((-12)²-4(1)(-380))]/2

 

x = (12±√1664)/2

x = (12 ±40.792)/2

x = 6 ± 20.396

Discarding the negative:

x = 26.396 cm length

x-12 = 14.396 cm width

height = 5 cm

 

volume:  LWH = 26.396(14.396)(5) = 1899.98 ≅ 1900 cm³

The small discrepancy is due to rounding errors

Answer 2

Let:

W = original width of cardboard (cm)

L = original length of cardboard (cm) = 12 + W

 

h = height of box = 5 cm

l = length of box (cm) = L - 2*h 

w = width of box = W - 2*h

 

v =  volume of box (cm^3) = l*w*h = 1900

 

v = (L-2*h)(W-2*h)*h   =  L*W*h - 2*L*h^2 - 2*W*h^2 + 4*h^3  = 1900  

 

h = 5 cm, insert into equation,  Note: h^2 = (5 cm)^2 = 25 cm^2  and 2*25 cm^2 = 50 cm^2

and 4*h^3 = 4*125 cm^3 = 500 cm^3 , Insert with results below:

 

(5 cm)*L*W - (50 cm^2)*L - (50 cm^2)*W + 500 cm^3 = 1900 cm^3        

 

or  simplifying:  (5 cm)*L*W - (50 cm^2)*(L + W) = 1400 cm^3                 

 

or w/o units:  5*L*W - 50*(L+W) = 1400                              mathematical statement as volume

 

To solve for unknown L and W, 

 

Simplify by dividing through each term by 5 cm:

 

L*W - (10 cm)*(L + W) = 280 cm^2                                mathematical statement using 2 unknown variable

 

or w/o units"   L*W - 10*(L+W) = 280

 

To find the original cardboard length (L) and width (W):

 

L = 12 cm + W

 

Substitute L = 12 cm +W and solve for W:

 

(12 cm + W)*W - (10 cm)*[(12 cm + W) + W] = 280 cm^2

 

 

(12 cm)*W + W^2  - 120 cm^2 + (20 cm)*W = 280 cm^2

 

(12 cm)*W + (20 cm)*W  + W^2 = 280 cm^2 + 120 cm^2 = 400 cm^2

 

simplify and rearrange as standard quadratic equation and solve:

 

W^2 + (32 cm)*W  - 400 cm^2 = 0          where: [ -b +/- √(b^2 - 4*a*c)]/2*a       a = 1    b = 32  c = -400

 

therefore, only use positive root:   W = [-32 + √( 32^2 - 4*1*(-400) )/2*1  = 9.612 cm

 

L = 12 cm + W = 12 cm + 9.612 cm = 21.612 cm