A 6-metre length of wire is cut into 12 pieces (calculus needed)

Question

A 6-metre length of wire is cut into 12 pieces, and these are used to construct a box on a square base. Show that each piece should be 50 cm for the box to have (a) maximum surface area, and (b) maximum volume

Answer 1

Let

x = side of each square base in cm

y = height of the box in cm

then

8x + 4 y =600 ⇒ y = 150 - 2x

 

a) The surface area of the box is

S = 4xy + 2x²

   = 4x(150 - 2x) + 2x²

   = 600x - 6x²

dS/dx = 600 - 12x

dS/dx = 0 ⇒ x = 50 cm, and y = 50 cm.

Note: x = 50 cm gives maximum surface area since dS/dx > 0 for x <50, and dS/dx < 0 for x >50.

 

b) The volume of the box

V = x²y

   = x²(150 - 2x)

   = 150x² - 2x³

dV/dx = 300x - 6x²

         = 6x(50 -x)

 

dV/dx = 0

⇒ x = 0  Discarded since no box can be made with size = 0.

   x = 50 and y = 50 cm

This gives maximum volume since dV/dx > 0 for x <50, and dV/dx < 0 for x >50.

Answer 2

The 3 lengths are a,b, and c.

 

Uncut length is 6 m.

 

4a + 4b + 4c = 6

 

(i)  a + b + c = 6/4 = 1.5 m

 

(ii)  A = 2(ab + bc + ac)

 

(iii)  V = abc

 

Square base implies 2 sides are equal.  Let a = b.

 

(i) becomes

 

2a + c = 1.5

c = 1.5 - 2a

 

(ii) becomes 

 

A = 2(a² + 2ac) = 2[a² + 2a(1.5 - 2a)] = 2a² + 6a - 8a² = 6a - 6a²

 

(iii) becomes 

 

V = a²(1.5 - 2a) = 1.5a² - 2a³

 

Maximum area:

 

A = 6a - 6a²

 

dA/da = 6 - 12a = 0 

 

a = 6/12 = 0.5 m = 50 cm

 

b = a = 50 cm

 

c = 1.5 - 2a = [1.5 - 2(0.5 m)] = (1.5 - 1.0) m = 0.5 m = 50 cm

 

Maximum volume:

 

V = 1.5a² - 2a³

 

dV/da = 3a - 6a² = 0

 

1 - 2a = 0

 

a = 0.5 m = 50 cm