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Date May 2014 Marks available 2 Reference code 14M.2.sl.TZ2.5
Level SL only Paper 2 Time zone TZ2
Command term Draw Question number 5 Adapted from N/A

Question

A parcel is in the shape of a rectangular prism, as shown in the diagram. It has a length l cm, width w cm and height of 20 cm.

The total volume of the parcel is 3000 cm3.

Express the volume of the parcel in terms of l and w.

[1]
a.

Show that l=150w.

[2]
b.

The parcel is tied up using a length of string that fits exactly around the parcel, as shown in the following diagram.



Show that the length of string, S cm, required to tie up the parcel can be written as

S=40+4w+300w, 0<w

 

[2]
c.

The parcel is tied up using a length of string that fits exactly around the parcel, as shown in the following diagram.



Draw the graph of S for 0 < w \leqslant 20 and 0 < S \leqslant 500, clearly showing the local minimum point. Use a scale of 2 cm to represent 5 units on the horizontal axis w (cm), and a scale of 2 cm to represent 100 units on the vertical axis S (cm).

[2]
d.

The parcel is tied up using a length of string that fits exactly around the parcel, as shown in the following diagram.



Find \frac{{{\text{d}}S}}{{{\text{d}}w}}.

[3]
e.

The parcel is tied up using a length of string that fits exactly around the parcel, as shown in the following diagram.



Find the value of w for which S is a minimum.

[2]
f.

The parcel is tied up using a length of string that fits exactly around the parcel, as shown in the following diagram.



Write down the value, l, of the parcel for which the length of string is a minimum.

[1]
g.

The parcel is tied up using a length of string that fits exactly around the parcel, as shown in the following diagram.



Find the minimum length of string required to tie up the parcel.

[2]
h.

Markscheme

20lw   OR   V = 20lw     (A1)

[1 mark]

a.

3000 = 20lw     (M1)

 

Note: Award (M1) for equating their answer to part (a) to 3000.

 

l = \frac{{3000}}{{20w}}     (M1)

 

Note: Award (M1) for rearranging equation to make l subject of the formula. The above equation must be seen to award (M1).

 

OR

150 = lw     (M1)

 

Note: Award (M1) for division by 20 on both sides. The above equation must be seen to award (M1).

 

l = \frac{{150}}{w}     (AG)

[2 marks]

b.

S = 2l + 4w + 2(20)     (M1)

 

Note: Award (M1) for setting up a correct expression for S.

 

2\left( {\frac{{150}}{w}} \right) + 4w + 2(20)     (M1)

 

Notes: Award (M1) for correct substitution into the expression for S. The above expression must be seen to award (M1).

 

= 40 + 4w + \frac{{300}}{w}     (AG)

[2 marks]

c.


    
 (A1)(A1)(A1)(A1)

 

Note: Award (A1) for correct scales, window and labels on axes, (A1) for approximately correct shape, (A1) for minimum point in approximately correct position, (A1) for asymptotic behaviour at w = 0.

     Axes must be drawn with a ruler and labeled w and S.

     For a smooth curve (with approximately correct shape) there should be one continuous thin line, no part of which is straight and no (one-to-many) mappings of w.

     The S-axis must be an asymptote. The curve must not touch the S-axis nor must the curve approach the asymptote then deviate away later.

 

[4 marks]

d.

4 - \frac{{300}}{{{w^2}}}     (A1)(A1)(A1)

 

Notes: Award (A1) for 4, (A1) for -300, (A1) for \frac{1}{{{w^2}}} or {w^{ - 2}}. If extra terms present, award at most (A1)(A1)(A0).

 

[3 marks]

e.

4 - \frac{{300}}{{{w^2}}} = 0   OR   \frac{{300}}{{{w^2}}} = 4   OR   \frac{{{\text{d}}S}}{{{\text{d}}w}} = 0     (M1)

 

Note: Award (M1) for equating their derivative to zero.

 

w = 8.66{\text{ }}\left( {\sqrt {75} ,{\text{ 8.66025}} \ldots } \right)     (A1)(ft)(G2)

 

Note: Follow through from their answer to part (e).

 

[2 marks]

f.

17.3 \left( {\frac{{150}}{{\sqrt {75} }},{\text{ 17.3205}} \ldots } \right)     (A1)(ft)

 

Note: Follow through from their answer to part (f).

 

[1 mark]

g.

40 + 4\sqrt {75}  + \frac{{300}}{{\sqrt {75} }}     (M1)

 

Note: Award (M1) for substitution of their answer to part (f) into the expression for S.

 

= 110{\text{ (cm) }}\left( {40 + 40\sqrt 3 ,{\text{ 109.282}} \ldots } \right)     (A1)(ft)(G2)

 

Note: Do not accept 109.

     Follow through from their answers to parts (f) and (g).

 

[2 marks]

h.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.
[N/A]
d.
[N/A]
e.
[N/A]
f.
[N/A]
g.
[N/A]
h.

Syllabus sections

Topic 6 - Mathematical models » 6.5 » Models using functions of the form f\left( x \right) = a{x^m} + b{x^n} + \ldots ; m,n \in \mathbb{Z} .
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