User interface language: English | Español

Date May 2022 Marks available 2 Reference code 22M.2.SL.TZ2.2
Level Standard Level Paper Paper 2 Time zone Time zone 2
Command term Calculate Question number 2 Adapted from N/A

Question

Scott purchases food for his dog in large bags and feeds the dog the same amount of dog food each day. The amount of dog food left in the bag at the end of each day can be modelled by an arithmetic sequence.

On a particular day, Scott opened a new bag of dog food and fed his dog. By the end of the third day there were 115.5 cups of dog food remaining in the bag and at the end of the eighth day there were 108 cups of dog food remaining in the bag.

Find the number of cups of dog food

In 2021, Scott spent $625 on dog food. Scott expects that the amount he spends on dog food will increase at an annual rate of 6.4%.

fed to the dog per day.

[3]
a.i.

remaining in the bag at the end of the first day.

[1]
a.ii.

Calculate the number of days that Scott can feed his dog with one bag of food.

[2]
b.

Determine the amount that Scott expects to spend on dog food in 2025. Round your answer to the nearest dollar.

[3]
c.

Calculate the value of Σn=110625×1.064n-1.

[1]
d.i.

Describe what the value in part (d)(i) represents in this context.

[2]
d.ii.

Comment on the appropriateness of modelling this scenario with a geometric sequence.

[1]
e.

Markscheme

EITHER

115.5=u1+3-1×d   115.5=u1+2d

108=u1+8-1×d   108=u1+7d         (M1)(A1)


Note: Award M1 for attempting to use the arithmetic sequence term formula, A1 for both equations correct. Working for M1 and A1 can be found in parts (i) or (ii).


d=-1.5

1.5 (cups/day)         A1


Note: Answer must be written as a positive value to award A1.


OR

d= 115.5-1085         (M1)(A1)


Note: Award M1 for attempting a calculation using the difference between term 3 and term 8; A1 for a correct substitution.


d= 1.5 (cups/day)         A1

 

[3 marks]

a.i.

u1= 118.5 (cups)         A1

 

[1 mark]

a.ii.

attempting to substitute their values into the term formula for arithmetic sequence equated to zero        (M1)

0=118.5+n-1×-1.5

n= 80 days        A1


Note: Follow through from part (a) only if their answer is positive.


[2 marks]

b.

t5=  625×1.0645-1       (M1)(A1)


Note: Award M1 for attempting to use the geometric sequence term formula; A1 for a correct substitution

 

$801       A1


Note: The answer must be rounded to a whole number to award the final A1.


[3 marks]

c.

S10=  $ 8390  8394.39         A1


[1 mark]

d.i.

EITHER

the total cost (of dog food)         R1

for 10 years beginning in 2021   OR  10 years before 2031         R1


OR

the total cost (of dog food)         R1

from 2021 to 2030 (inclusive)  OR  from 2021 to (the start of) 2031         R1


[2 marks]

d.ii.

EITHER
According to the model, the cost of dog food per year will eventually be too high to keep a dog.

OR
The model does not necessarily consider changes in inflation rate.

OR
The model is appropriate as long as inflation increases at a similar rate.

OR
The model does not account for changes in the amount of food the dog eats as it ages/becomes ill/stops growing.

OR
The model is appropriate since dog food bags can only be bought in discrete quantities.        R1


Note: Accept reasonable answers commenting on the appropriateness of the model for the specific scenario. There should be a reference to the given context. A reference to the geometric model must be clear: either “model” is mentioned specifically, or other mathematical terms such as “increasing” or “discrete quantities” are seen. Do not accept a contextual argument in isolation, e.g. “The dog will eventually die”.


[1 mark]

e.

Examiners report

Parts (a) and (b) were mostly well answered, but some candidates ignored the context and did not give the number of dog food cups per day as a positive number. Most candidates considered geometric sequence in part (c) correctly, and used the correct formula for the nth term, although they used an incorrect value for n at times. Some candidates used the finance application incorrectly. The sum in part (d) was calculated correctly by some candidates, although many seemed unfamiliar with sigma notation and with calculating summations using GDC. In part (d), most candidates interpreted correctly that the sum represented the cost of dog food for 10 years but did not identify the specific 10-year period. Part (e) was not answered well – often candidates made very general and abstract statements devoid of any contextual references.

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

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series
Show 111 related questions
Topic 1—Number and algebra

View options