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Date May Specimen paper Marks available 2 Reference code SPM.2.SL.TZ0.4
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Calculate Question number 4 Adapted from N/A

Question

The Happy Straw Company manufactures drinking straws.

The straws are packaged in small closed rectangular boxes, each with length 8 cm, width 4 cm and height 3 cm. The information is shown in the diagram.

Each week, the Happy Straw Company sells x boxes of straws. It is known that dPdx=2x+220, x ≥ 0, where P is the weekly profit, in dollars, from the sale of x thousand boxes.

Calculate the surface area of the box in cm2.

[2]
a.

Calculate the length AG.

[2]
b.

Find the number of boxes that should be sold each week to maximize the profit.

[3]
c.

Find P(x).

[5]
d.

Find the least number of boxes which must be sold each week in order to make a profit.

[3]
e.

Markscheme

2(8 × 4 + 3 × 4 + 3 × 8)        M1

= 136 (cm2)        A1

[2 marks]

a.

82+42+32        M1

(AG =) 9.43 (cm) (9.4339…, 89)        A1

[2 marks]

b.

2x+220=0       M1

x=110        A1

110 000 (boxes)        A1

[3 marks]

c.

P(x)=2x+220dx      M1

Note: Award M1 for evidence of integration.

P(x)=x2+220x+c       A1A1

Note: Award A1 for either x2 or 220x award A1 for both correct terms and constant of integration.

1700=(20)2+220(20)+c       M1

c=2300

P(x)=x2+220x2300      A1

[5 marks]

d.

x2+220x2300=0     M1

x=11.005       A1

11 006 (boxes)      A1

Note: Award M1 for their P(x)=0, award A1 for their correct solution to x.
Award the final A1 for expressing their solution to the minimum number of boxes. Do not accept 11 005, the nearest integer, nor 11 000, the answer expressed to 3 significant figures, as these will not satisfy the demand of the question.

[3 marks]

e.

Examiners report

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

Syllabus sections

Topic 5—Calculus » SL 5.3—Differentiating polynomials, n E Z
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