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Integration and Its Applications · Paper 1 — Free Maths AI SL Practice Questions

1FoundationSAQ-SIndefinite integrals and their properties7 marksPaper 1~11 min

The rate at which water flows into a reservoir is modelled by

f(t) = 12t^2 + 5

, where

f(t)

is measured in thousands of litres per day and

t

is the time in days from the start of monitoring.

(a)

Find an expression for

F(t)

, the total amount of water, in thousands of litres, that has flowed into the reservoir after

t

days. [2 marks]

(b)

Find the total amount of water that flows into the reservoir during the first 3 days. [2 marks]

(c)

The reservoir has a capacity of

500

thousand litres and contains

250

thousand litres at

t = 0

. Determine the day on which the reservoir becomes full, justifying your answer. [3 marks]

Solutions

2FoundationSAQ-SIndefinite integrals and their properties5 marksPaper 1~8 min

The marginal cost of producing

x

units of a product is given by

C'(x) = 3x^2 - 10x + 20

dollars per unit. The fixed costs are

500

dollars (i.e.

C(0) = 500

(a)

Find an expression for the total cost function

C(x)

. [2 marks]

(b)

The average cost per unit is defined as

A(x) = \dfrac{C(x)}{x}

for

x > 0

. Find the value of

x

that minimises

A(x)

, and show that is a minimum. [3 marks]

Solutions

3MasterySAQ-SIndefinite integrals and their properties7 marksPaper 1~11 min

The rate of change of the volume of water,

V

(in litres), in a tank during a 10-minute period is given by

\dfrac{dV}{dt} = 12t - 3t^2

, where

t

is the time in minutes,

0 \leq t \leq 10

(a)

Show that

V(t) = 6t^2 - t^3 + C

, where

C

is a constant. [2 marks]

(b)

Given that the tank initially contains

50

litres of water, write down an expression for

V(t)

. [1 mark]

(c)

Find the volume of water in the tank after

5

minutes. [2 marks]

(d)

Determine whether the volume of water is increasing or decreasing at

t = 5

minutes. Justify your answer. [2 marks]

Solutions

4ChallengeSAQ-LIndefinite integrals and their properties8 marksPaper 1~12 min

A company manufactures a new type of solar panel. The rate of change of energy output, in kilowatt-hours per day, of a prototype panel is modelled by the function

R(t) = 12\text{e}^{-0.2t} + 4\cos\left(\frac{\pi t}{12}\right)

where

t

is the time in days since the panel was first activated, for

t \geq 0

. The total energy output

E(t)

, in kilowatt-hours, satisfies

E(t) = \int R(t)\,dt

and

E(0) = 0

(a)

Determine an expression for

E(t)

in the form

E(t) = A\text{e}^{-0.2t} + B\sin\!\left(\frac{\pi t}{12}\right) + C

where

A

B

, and

C

are constants. [4 marks]

(b)

Calculate the total energy output after

30

days, correct to three significant figures. [2 marks]

(c)

The company claims that the panel produces more energy in the second

15

days (days

15

30

) than in the first

15

days (days

0

15

). Determine whether this claim is supported by the model, justifying your answer with calculations. [2 marks]

Solutions

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