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Modeling with Functions · Paper 1 — Free Maths AA SL Practice Questions

1FoundationSAQ-SReal-world applications of functions (e.g. growth models)s7 marksPaper 1~11 min
A small business purchases a delivery van for 28 00028\,000 dollars. The value of the van depreciates linearly each year. After 55 years, the van is valued at 13 00013\,000 dollars.
(a)
Calculate the annual depreciation of the van in dollars per year. [2 marks]
(b)
Find the value of the van after 88 years. [2 marks]
(c)
The business plans to sell the van when its value first reaches 1 0001\,000 dollars. Determine the year in which this occurs and evaluate whether the linear model is appropriate for this prediction. [3 marks]
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2FoundationSAQ-SReal-world applications of functions (e.g. growth models)s5 marksPaper 1~8 min
The height of a plant is recorded weekly. At week 11, the plant is 4 cm4\,\text{cm} tall. At week 44, the plant is 13 cm13\,\text{cm} tall. The growth is assumed to be linear.
(a)
Find the linear function H(w)=mw+cH(w) = mw + c that gives the height HH in cm after ww weeks. [3 marks]
(b)
A second plant grows according to G(w)=5w−3G(w) = 5w - 3. Determine the week at which both plants are the same height, and state which plant is taller at week 1010. [2 marks]
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3MasterySAQ-SReal-world applications of functions (e.g. growth models)s5 marksPaper 1~8 min
The population of a small town is modelled by P(t)=50 000+1200tP(t) = 50\,000 + 1200t, where PP is the population and tt is the number of years after 1 January 2020.
(a)
Show that P(t+1)−P(t)=1200P(t+1) - P(t) = 1200 for all values of tt. [2 marks]
(b)
Find the value of tt for which the population first exceeds 65 00065\,000. [2 marks]
(c)
A government grant is available only towns whose population remains below 80 00080\,000 throughout the entire period 0≤t≤200 \leq t \leq 20. Determine, with justification, whether this town is eligible for the grant. [1 mark]
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4ChallengeSAQ-LReal-world applications of functions (e.g. growth models)s8 marksPaper 1~12 min
A company produces solar panels. The total revenue RR (in thousands of dollars) from selling xx thousand panels is given by R(x)=120x−3x2R(x) = 120x - 3x^2, for 0≤x≤400 \leq x \leq 40. The total cost CC (in thousands of dollars) is given by C(x)=20x+400C(x) = 20x + 400.
(a)
Show that the profit function is P(x)=−3x2+100x−400P(x) = -3x^2 + 100x - 400. [2 marks]
(b)
Calculate the value of xx that maximises P(x)P(x), and the corresponding maximum profit. [3 marks]
(c)
For x>40x > 40, the revenue function changes to R2(x)=80x−x2R_2(x) = 80x - x^2, while C(x)C(x) remains the same. Evaluate whether the company can achieve a profit greater than the maximum found in part (b) for any x>40x > 40. Justify your answer. [3 marks]
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Solutions

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