Practise questions · Salam Academy

SA-9709-P3-DE-0041 markL1 FoundationCalculatorForming

The temperature

\theta

of a cooling liquid decreases at a rate proportional to the amount by which it exceeds the room temperature of

20^\circ

C. With

k>0

, which equation models this?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE-0101 markL1 FoundationCalculatorForming

The volume

V

of a balloon increases at a rate inversely proportional to

\sqrt V

. Which equation models this?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0011 markL1 FoundationCalculatorForming

The mass

m

of a melting block of ice decreases at a rate proportional to the square of

m

. Which equation models this, with

k>0

?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0022 marksL1 FoundationCalculatorRates and constants

A population

N

satisfies

\dfrac{dN}{dt}=kN

. When

N=200

, the population is increasing at

10

per year. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0081 markL1 FoundationCalculatorForming

The function

y=5e^{-2t}

satisfies

\dfrac{dy}{dt}=ky

. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-G011 markL1 FoundationCalculatorForming

The area

A

of a patch of mould increases at a rate proportional to

A

. Which equation models this, with

k>0

?

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-Q011 markL1 FoundationCalculatorForming

The pressure

P

changes at a rate inversely proportional to

t^2

. This means

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-Q021 markL1 FoundationCalculatorRates and constants

At a certain time,

\dfrac{dN}{dt}=-40

. This tells you that at that time

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0032 marksL2 StandardCalculatorRates and constants

A liquid cools according to

\dfrac{d\theta}{dt}=-k(\theta-20)

,

k>0

. When

\theta=80

, the temperature is falling at

3^\circ

C per minute. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0042 marksL2 StandardCalculatorRates and constants

The depth

h

cm of water in a tank satisfies

\dfrac{dh}{dt}=-k\sqrt h

. When

h=25

the depth is decreasing at

0.5

cm per minute. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0053 marksL2 StandardCalculatorRates and constants

The volume of a cube of side

x

cm increases at a constant rate of

6

cm

^3

s

^{-1}

. Find the rate of increase of

x

when

x=2

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0072 marksL2 StandardCalculatorRates and constants

The number

x

of people who have heard a rumour satisfies

\dfrac{dx}{dt}=kx(10-x)

, where

x

is in thousands. When

x=2

, the rumour is spreading at

4

thousand people per day. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0063 marksL3 ExamCalculatorRates and constants

The area of a circular oil slick increases at

10

m

^2

per minute. Find the exact rate at which the radius is increasing when the radius is

5

m.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P3-DE1-0094 marksL4 ChallengingCalculatorForming

Water is poured into a tank at

20

cm

^3

s

^{-1}

. It leaks out at a rate proportional to the volume

V

cm

^3

in the tank, with constant of proportionality

0.1

. Form a differential equation for

V

. Find the volume at which

V

stops changing, and the rate of change of

V

when

V=50

.

Salam Academy Original Practice · Proposed mark scheme

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