6.3 - 2014 Q10
Figure 4 shows a closed letter box 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 cm2.
The cross section ABCD of the letter box is a trapezium with edges of lengths DA = 9x cm, AB = 4x cm, BC = 6x cm and CD = 5x cm as shown in Figure 5. The angle DAB = 90° and the angle ABC = 90°.
The volume of the letter box is 9600 cm3.
(a) Show that: (2)
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 cm2.
The cross section ABCD of the letter box is a trapezium with edges of lengths DA = 9x cm, AB = 4x cm, BC = 6x cm and CD = 5x cm as shown in Figure 5. The angle DAB = 90° and the angle ABC = 90°.
The volume of the letter box is 9600 cm3.
(a) Show that: (2)
(b) Hence show that the surface area of the letter box, S cm2, is given by: (4)
(c) Use calculus to find the minimum value of S. (6)
(d) Justify, by further differentiation, that the value of S you have found is a minimum. (2)
(d) Justify, by further differentiation, that the value of S you have found is a minimum. (2)