Differentiation

Edexcel IAL WMA12/01 /P2/Jan 2023/Q2 (Differentiation, Application, Optimization)

In this question you must show all stages of your working.​​ 

Solutions based entirely on calculator technology are not acceptable.

A brick is in the shape of a cuboid with width x cm, length 3x cm and height hcm, as shown in Figure 2.​​ 

The volume of the brick is 972cm3

• Show that the surface area of the brick, S cm2, is given by​​ 

S=6x2+2592x

(3)

• Find​​ dSdx

(1)

• Hence find the value of x for which S is stationary.​​ 

(2)

• Find​​ d2Sdx2​​ and hence show that the value of x found in part (c) gives the minimum value of S.​​ 

(2)

• Hence find the minimum surface area of the brick.​​ 

(1)

SOLUTION​​ 

a- Use the formula of surface area to show the given expression.​​ 

(Total Surface Area is the sum of the areas of all the faces of the prism. In this case, cuboid have 6 faces. The area of side​​ faces is​​ 2xh;​​ top and bottom face is​​ 23xh, and front and back face is​​ 23xh).​​ 

S=2xh+23xh+23xh

S=2 x h+6 x h+6x2

S=6x2+8 x h

(The given surface area expression that needs to be shown has no​​ h​​ parameter in it, so use the given information of volume to substitute the value of h in terms of x in the above expression to show the asked expression.) ​​ 

V=w x l x h

x 3xh+972

3x2h=972

(Now, substituting the value of h in terms of x in the expression derived of surface area.)

S=6x2+8x 324x2

h=9723x2

h=324x2

S=6x2+2592x   

b- ​​ To find​​ dsdx,​​ simply differentiate the surface area expression once.​​ 

S=6x2+2592x

 S= 6x2+2592 x-1

dsdx=12x-2592 x-2  

c- To find the value of x for which S is stationary, equate​​ dsdx​​ to zero.​​ 

(Remember, to find the x-cordinate of a stationary point of the curve, simply differentiate the equation of the curve and let it be equal to zero because at stationary points, whether it is local maxima or local minima, the gradient or​​ dydx is always zero at that stationary point.)

dsdx=0

12x-2592x2= 0

12x3-2592=0

12x3=2592

x3=259212

x= 2592123

(Choose only positive value of x as dimensions of the cuboid cant be negative.)

x=6cm

d- Differentiating the first derivative of​​ S​​ to find double derivative and then putting the value of x as​​ 6​​ in the double derivated ex[pression to find the nature of S at​​ x=6.

(If a function​​ f(x)​​ has a stationary point when​​ x=a, then if​​ f’’(a)>0, the point is a local minimum)​​ dsdx=12x-2592 x-2

d2sdx2=12+ 5184 x-3

x=6⇒d2sdx=12+518463=36

As d2sdx2>0

Hence,​​ x=6 gives the minimum value of S.​​ 

e-​​ Substituting the value of​​ x as 6, to get minimum value of S.

S=  662+25926

S=648 cm2