**6.3 - 2014 Q10**

Figure 4 shows a closed letter box

The letter box is a right prism of length

The cross section

The volume of the letter box is 9600 cm

(a) Show that: (2)

*ABFEHGCD*, which is made to be attached to a wall of a house.The letter box is a right prism of length

*y*cm as shown in Figure 4. The base*ABFE*of the prism is a rectangle. The total surface area of the six faces of the prism is*S*cm^{2}.The cross section

*ABCD*of the letter box is a trapezium with edges of lengths*DA*= 9*x*cm,*AB*= 4*x*cm,*BC*= 6*x*cm and*CD*= 5*x*cm as shown in Figure 5. The angle*DAB*= 90° and the angle*ABC*= 90°.The volume of the letter box is 9600 cm

^{3}.(a) Show that: (2)

(b) Hence show that the surface area of the letter box,

*S*cm^{2}, is given by: (4)(c) Use calculus to find the minimum value of

(d) Justify, by further differentiation, that the value of

*S*. (6)(d) Justify, by further differentiation, that the value of

*S*you have found is a minimum. (2)