Solved: Scott purchases food for his dog in large bags and feeds the dog the same amount... [IB Mathematics Applications & Interpretation (Math AI)]

Question

SLPaper 2

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]

Verified

Solution

EITHER

$115.5 = u_{1} + (3 - 1) \times d (115.5 = u_{1} + 2 d)$

$108 = u_{1} + (8 - 1) \times d (108 = u_{1} + 7 d)$ (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.

$(d =) \frac{115.5 - 108}{5}$ (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]

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

[1]

Verified

Solution

$(u_{1} =) 118.5$ (cups) A1

[1 mark]

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

[2]

Verified

Solution

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

$0 = 118.5 + (n - 1) \times (- 1.5)$

$(n =) 80$ days A1

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

[2 marks]

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

[3]

Verified

Solution

$(t_{5} =) 625 \times 1 . 064^{(5 - 1)}$ (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]

Calculate the value of $Σ_{n = 1}^{10} (625 \times 1 . 064^{(n - 1)})$ .

[1]

Verified

Solution

$(S_{10} =) ($) 8390 (8394.39 \dots)$ A1

[1 mark]

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

[2]

Verified

Solution

EITHER

the total cost (of dog food) R1

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

the total cost (of dog food) R1

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

[2 marks]

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

[1]

Verified

Solution

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]

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