You're viewing 3 free preview questions for Maths AA HL — subscribe to unlock 124 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

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

1FoundationSAQ-SReal-world applications of functions (e.g. growth models)s8 marksPaper 1~12 min

Criterion: Find Marks: 8 A small town has a population of

12\,500

people. A demographer models the population

t

years from now using the linear function

P(t) = 12500 + 340t

(a)

Write down the values of the initial population and the annual rate of change, and state what each represents in context. [2 marks]

(b)

Determine the number of complete years it will take for the population to first exceed

15\,000

people. [2 marks]

(c)

The demographer notes that a neighbouring town has population

Q(t) = 18000 - 120t

. Determine the year in which the two towns have equal populations, and evaluate whether a linear model is likely to remain valid for both towns beyond this point. [4 marks]

Solutions

2MasterySAQ-SReal-world applications of functions (e.g. growth models)s5 marksPaper 1~8 min

A biologist is studying the population of a species of bird on a remote island. The population size

P

(in hundreds) is modelled by

P(t) = 12 + 3\ln(t+1)

, where

t

is the time in years since the study began.

(a)

Show that the population first reaches 2700 birds after

T = e^5 - 1

years. [2 marks]

(b)

Find the average rate of change of

P

, in hundreds of birds per year, over the interval

0 \leq t \leq T

, giving your answer in exact form. [2 marks]

(c)

The biologist claims that the rate of growth of the population is always decreasing. Determine whether this claim is correct, justifying your answer with a mathematical argument. [1 mark]

Solutions

3ChallengeSAQ-LReal-world applications of functions (e.g. growth models)s8 marksPaper 1~12 min

A pharmaceutical company is testing a new drug. The concentration

C(t)

(in mg/L) of the drug in a patient's bloodstream at time

t

(in hours) after administration is modelled by

C(t) = \frac{at}{t^2 + b}, \quad t \geq 0,

where

a

and

b

are positive constants.

(a)

Calculate the value of

t

at which the maximum concentration occurs, expressing your answer in terms of

b

. [3 marks]

(b)

The maximum concentration is

C_{\max} = 5\,\text{mg/L}

, occurring at

t = 2

hours. Determine the values of

a

and

b

. [3 marks]

(c)

A drug is considered therapeutically effective while

C(t) \geq 1\,\text{mg/L}

. Using your values from (b), determine the total duration for which the drug remains therapeutically effective. [2 marks]

Solutions

Questions 4-127

Unlock these questions with Premium

That's every free question here — 3 of 127.

+124 more questions, plus Mock Papers & Worksheets, are waiting in Premium.

Upgrade