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.
Calculate the length AG.
Find the number of boxes that should be sold each week to maximize the profit.
Find P(x).
Find the least number of boxes which must be sold each week in order to make a profit.
Markscheme
2(8 × 4 + 3 × 4 + 3 × 8) M1
= 136 (cm2) A1
[2 marks]
√82+42+32 M1
(AG =) 9.43 (cm) (9.4339…, √89) A1
[2 marks]
−2x+220=0 M1
x=110 A1
110 000 (boxes) A1
[3 marks]
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+220x−2300 A1
[5 marks]
−x2+220x−2300=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]