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Date November 2018 Marks available 2 Reference code 18N.2.SL.TZ0.T_6
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Calculate Question number T_6 Adapted from N/A

Question

Haruka has an eco-friendly bag in the shape of a cuboid with width 12 cm, length 36 cm and height of 9 cm. The bag is made from five rectangular pieces of cloth and is open at the top.

 

Nanako decides to make her own eco-friendly bag in the shape of a cuboid such that the surface area is minimized.

The width of Nanako’s bag is x cm, its length is three times its width and its height is y cm.

 

The volume of Nanako’s bag is 3888 cm3.

Calculate the area of cloth, in cm2, needed to make Haruka’s bag.

[2]
a.

Calculate the volume, in cm3, of the bag.

[2]
b.

Use this value to write down, and simplify, the equation in x and y for the volume of Nanako’s bag.

[2]
c.

Write down and simplify an expression in x and y for the area of cloth, A, used to make Nanako’s bag.

[2]
d.

Use your answers to parts (c) and (d) to show that

A = 3 x 2 + 10368 x .

[2]
e.

Find d A d x .

[3]
f.

Use your answer to part (f) to show that the width of Nanako’s bag is 12 cm.

[3]
g.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

36 × 12 + 2(9 ×12) + 2(9 × 36)      (M1)

Note: Award (M1) for correct substitution into surface area of cuboid formula.

 

= 1300 (cm2)  (1296 (cm2))       (A1)(G2)

 

[2 marks]

a.

36 × 9 ×12     (M1)

Note: Award (M1) for correct substitution into volume of cuboid formula.

 

= 3890 (cm3)  (3888 (cm3))       (A1)(G2)

 

[2 marks]

b.

3 x  × x  × y  = 3888    (M1)

Note: Award (M1) for correct substitution into volume of cuboid formula and equated to 3888.

 

x 2 y  = 1296      (A1)(G2)

Note: Award (A1) for correct fully simplified volume of cuboid.

Accept y = 1296 x 2 .

 

[2 marks]

c.

(A =) 3x2 + 2(xy) + 2(3xy)    (M1)

Note: Award (M1) for correct substitution into surface area of cuboid formula.

 

(A =) 3x2 + 8xy       (A1)(G2)

Note: Award (A1) for correct simplified surface area of cuboid formula.

 

 

[2 marks]

d.

A = 3 x 2 + 8 x ( 1296 x 2 )      (A1)(ft)(M1)

Note: Award (A1)(ft) for correct rearrangement of their part (c) seen (rearrangement may be seen in part(c)), award (M1) for substitution of their part (c) into their part (d) but only if this leads to the given answer, which must be shown.

 

A = 3 x 2 + 10368 x      (AG) 

 

[2 marks]

e.

( d A d x ) = 6 x 10368 x 2       (A1)(A1)(A1)

Note: Award (A1) for 6 x , (A1) for −10368, (A1) for x 2 . Award a maximum of (A1)(A1)(A0) if any extra terms seen.

 

[3 marks]

f.

6 x 10368 x 2 = 0         (M1)

Note: Award (M1) for equating their  d A d x  to zero.

 

6 x 3 = 10368   OR   6 x 3 10368 = 0    OR    x 3 1728 = 0         (M1)

Note: Award (M1) for correctly rearranging their equation so that fractions are removed.

 

x = 1728 3         (A1)

x = 12  (cm)       (AG)

Note: The (AG) line must be seen for the final (A1) to be awarded. Substituting x = 12 invalidates the method, award a maximum of (M1)(M0)(A0).

 

[3 marks]

g.

Examiners report

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

Syllabus sections

Topic 5 —Calculus » SL 5.1—Introduction of differential calculus
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Topic 5 —Calculus » SL 5.3—Differentiating polynomials, n E Z
Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion
Topic 5 —Calculus

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