Exponential Functions (AQA Level 3 Mathematical Studies (Core Maths): Paper 2C: Graphical Techniques): Exam Questions

An online shop opened on 1 January 2014

The number of customers using the shop each day, , increased exponentially between 2019 and 2022

An approximate value of can be found using

for

where is the number of years the shop has been open.

Estimate the number of customers using the shop on 1 July 2021

State your answer to two significant figures.

Explain how your answer to question 6(a) relates to the rate at which the number of customers is increasing on 1 July 2021

A study has shown that temperatures greater than 24°C can have an impact on students’ learning.

The study used a measure called ‘Ability to learn’, , which is measured as a percentage.

can be modelled by the equation

for

where is the temperature in degrees Celsius (°C) and is a constant.

Explain fully what the value 0.98 tells us about the value of as increases.

When = 24 , = 100

Work out the value of

= .......................................................

A different study led to the model

for

Use this model to work out the value of that gives = 75

= ........................................

Water hyacinths are plants that grow in lakes and cover the water’s surface.

When uncontrolled, the plants grow at an exponential rate.

The growth in one lake is recorded by measuring the surface area of water covered by the plants.

Initially the plants cover a surface area of 4.5m²

Measurements are taken weekly and a graph is plotted to model the data.

Estimate how many days after the first measurement the plants are growing at a rate of 1m² per day.

Answer .....................................................days

The data can also be modelled by an equation of the form

is the surface area covered in m²

is the days after the first measurement.

and are constants.

Work out the values of and

=.....................................................

= .....................................................

State two reasons why the model might not be suitable for predicting the surface area of water covered by the plants one year after the first measurement.

The temperature of a computer processor increases from the moment it is turned on.

The temperature is exponential and modelled by the equation

is the temperature of the processor (°C)

is the time in minutes after the computer is turned on.

Work out the temperature of the processor 30 seconds after being turned on.

Answer .......................................... °C

When the processor reaches 45°C, cooling fans start.

(i) Work out the time it takes for the cooling fans to start.

Give your answer in minutes and seconds.

Answer .......................... minutes ............................. seconds

[4]

(ii) Work out the rate at which the temperature is increasing just as the fans start.

State the units of your answer.

Answer ...................................................

[3]

The graph below models the processor temperature in °C from time minutes after the fans start.

Use the graph to estimate the rate of cooling when the processor returns to 18°C

Give your answer in degrees Celsius per second.

Answer ................................................°C per second

The intensity of direct sunlight can be measured in lumens.

Light intensity decreases in the hour before sunset.

At sunset there is zero direct sunlight. The equation to model the intensity of direct sunlight, lumens, at time minutes before sunset is

where

is the intensity of direct sunlight one hour before sunset.

The constant takes into consideration the atmospheric conditions, location and time of year.

On one particular day,

• the sun sets at 6 pm

• the light intensity 10 minutes before sunset is half the light intensity at 5 pm

• the light intensity 30 minutes before sunset is 85 000 lumens.

Calculate the light intensity predicted by the model 5 minutes before sunset.

Answer ................................... lumens

A social media post can very quickly receive large viewing figures.

The total number of views, , of one post minutes after being posted on social media is modelled by the equation

The model is only a good predictor of views for certain values of

(i) When might this model not be a good predictor for the total number of views?

Suggest a reason why.

[2]

(ii) Use the model to estimate the number of minutes it would take the post to reach one million total views.

Answer .....................................minutes

Two adverts are posted on social media at the same time.

They receive total number of views, minutes after being posted.

Advert A has linear growth, with total number of views modelled by the equation

Advert B has exponential growth, with total number of views modelled by the equation

Work out the value of m for which the total numbers of views of both adverts are predicted to have the same rate of change.

= ......................................................

An air-freshener sprays particles into the air.

The graph shows the concentration of particles in the air over time, measured in parts per million (ppm).

The concentration, ppm, at time minutes after the air-freshener is sprayed, is modelled by the equation

where and are constants.

Explain why must be negative.

Use the graph to work out the values of and .

= ....................................... = ...................................

The graph shows an example of exponential growth.

A student models this using the equation

State the value of the constant .

Answer...................................

What is the gradient of the curve when ?

Work out the value of when

Give your answer as a decimal to 3 decimal places.

Answer ............................................

Musical instruments are used to play notes which have different frequencies.

Each note has its own frequency, measured in hertz (Hz).

The frequencies of 13 consecutive notes on a piano can be modelled by the function , where takes integer values from 0 to 12

The frequencies of these notes are shown in the graph below.

The frequency of note 0 is given by = 262

The function is defined as

where is a constant.

State whether is positive or negative.

Give a reason for your answer.

The frequency of note 12 is double the frequency of note 0

Work out the value of to 2 decimal places.

Answer ...............................

Veronica posts an interesting video on Facebook.

The total number of views, , at time hours after the video was first viewed is modelled by

Work out the total number of views 15 hours after the video was first viewed.

Answer..........................

Work out the value of when the total number of views is 3000

Answer .................................

Work out the rate at which the number of views was increasing 3 hours after the video was first viewed.
You may use the table and the grid below.

Answer .....................................

Work out the time taken for the number of views to double.