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Mathematical Modelling · Guided Practice

Linear Relationships

9 questions with answers and video solutions. Try each one before revealing the answer.

1
A gym charges a joining fee of £30\displaystyle \pounds 30, plus £22\displaystyle \pounds 22 for each month of membership.
Write down a formula for the total cost C\displaystyle C, in pounds, after m\displaystyle m months.
Video solution coming soon
2
The total cost of gym membership is modelled by C=30+22m\displaystyle C = 30 + 22m, where m\displaystyle m is the number of months.
Calculate the total cost after 9\displaystyle 9 months.
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3
A tank contains 950\displaystyle 950 litres of water and drains at a constant rate of 35\displaystyle 35 litres per minute.
Write down a formula for the volume V\displaystyle V, in litres, remaining after t\displaystyle t minutes.
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4
The volume of water in a tank is modelled by V=950−35t\displaystyle V = 950 - 35t, where t\displaystyle t is the time in minutes.
Calculate the volume remaining after 12\displaystyle 12 minutes.
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5
The volume of water in a tank is modelled by V=950−35t\displaystyle V = 950 - 35t, where t\displaystyle t is the time in minutes.
State the gradient of this model and explain what it tells you in this context.
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6
One firm charges a call-out fee of £48\displaystyle \pounds 48 plus £26\displaystyle \pounds 26 per hour.
A rival charges a call-out fee of £75\displaystyle \pounds 75 plus £17\displaystyle \pounds 17 per hour.
Determine which firm is cheaper for a job lasting 4\displaystyle 4 hours. Use your working to justify your answer.
Video solution coming soon
7
A streaming service has 1800\displaystyle 1800 subscribers and gains 145\displaystyle 145 new subscribers each week.
(a) Write down a formula for the number of subscribers S\displaystyle S after w\displaystyle w weeks.
(b) Calculate the number of subscribers after 15\displaystyle 15 weeks.
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8
The number of items left in a warehouse is modelled by N=240−24d\displaystyle N = 240 - 24d, where d\displaystyle d is the number of days.
(a) State the number of items at the start.
(b) Calculate the number of items left after 6\displaystyle 6 days.
Video solution coming soon
9
2025 Q10
(3, 1, 1, 2)7 Marks

A cylindrical container is being filled with water.

Cylindrical container filling with water

Initially, the depth of water is 4 centimetres. The depth of the water in the container increases at a constant rate of 3 centimetres per minute. It takes 8 minutes to finish filling the container.

(a) Draw a graph to model how the depth of water in the container changes over time until it is filled.

Graph grid for filling a cylindrical container

(b)(i) State the type of relationship modelled in your graph.
(b)(ii) State the dependent variable in your graph.

(c) Another container is shown below. This container is also filled with water at a constant rate of 5 cubic centimetres per second.

Conical container

Three graphs are shown below showing how the depth of water in the container changes over time.

Three graphs: A (curving up), B (linear), C (curving down)

Explain which graph could model the depth of water in the container.

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