A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 5: Straight line graphs · Lesson 3

Modelling with straight lines

70 marks

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Quick questions to start.

A1
Needed today
Find an equation of the line through $(5, 9)$ and $(15, 33)$ in the form $y = m x + c$ .
[2 marks]
Answer $y = \dots\dots$
A2
Last lesson
Show that the lines $5 x - 2 y = 3$ and $4 x + 10 y = 1$ are perpendicular.
[2 marks]
A3
From chapter 3
Solve $12 + 0.8 t = 30 - 0.4 t$ .
[2 marks]
Answer $t = \dots\dots$
A4
GCSE Higher
$y$ is directly proportional to $x$ , and $y = 42$ when $x = 12$ . Find $y$ when $x = 30$ .
[2 marks]
Answer $y = \dots\dots$

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Tiles needed, $n$ , is directly proportional to floor area $A m^{2}$ , and $4.5 m^{2}$ needs $72$ tiles. Find $k$ in $n = k A$ , say what $k$ means, and find $n$ for $7.25 m^{2}$ .

Hint A straight line through the origin:

k = \frac{n}{A}

B1
Paint used, $V$ litres, is directly proportional to area $A m^{2}$ , and $12 m^{2}$ uses $1.8$ litres. Find $V = k A$ , say what $k$ means, and find the area $4.5$ litres covers.
[4 marks]
Answer

Example 2

A courier charges a fixed fee plus a rate per kg: $4$ kg costs $£11.20$ and $9$ kg costs $£17.70$ . Find $C = a w + b$ , say what $a$ and $b$ mean, and find the mass of a $£25.50$ parcel.

Hint The parcels are two points

(w, C)

on one line.

B2
Hall hire is a fixed fee plus a rate per hour: $3$ hours cost $£87$ and $7$ hours $£163$ . Find $C = ah + b$ , say what $a$ and $b$ mean, and find the hours in a $£353$ booking.
[5 marks]
Answer

Example 3

A stack of $n$ cups is $h$ cm tall: $n = 1, 2, 3, 4, 5$ give $h = 11.0, 12.4, 13.8, 15.2, 16.6$ . Explain why a linear model fits, find $h = an + b$ , and find how many cups fit under a $40$ cm shelf.

Hint Find the step in

h

each time

n

goes up by

1

B3
A stack of $n$ chairs is $h$ cm tall: $n = 2, 4, 6, 8$ give $h = 84, 96, 108, 120$ . Explain why a linear model fits, find $h = an + b$ , and find the most chairs in a stack under $190$ cm.
[4 marks]
Answer $h = \dots\dots$ $n = \dots\dots$

Practice

A-Level

C1
A cyclist starts $6$ km from home and rides straight away from home at $18$ km per hour. (a) Write a model for the distance $d$ km from home $t$ minutes later. (b) Find when the cyclist is $15$ km from home. (c) State one assumption of the model.
[4 marks]
Answer $d = \dots\dots$ $t = \dots\dots$

C2
Data set $P$ is $(0, 3.5)$ , $(2, 8.1)$ , $(4, 12.7)$ , $(6, 17.3)$ . Data set $Q$ is $(0, 2)$ , $(1, 5)$ , $(2, 10)$ , $(3, 17)$ . Which set fits a linear model? Find its equation.
[4 marks]
Answerset $\dots\dots$ $y = \dots\dots$
C3
The number of visitors $n$ to a café on a day is modelled by $n = 6.5 T - 48$ , where $T^{\circ} C$ is the temperature. (a) Say what $6.5$ means. (b) Explain why the model cannot be used when $T = 5$ .
[3 marks]
C4
A machine is bought for $£14 400$ and loses $£1650$ in value each year. Write a linear model for its value $V$ after $t$ years, find after how many complete years its value is first below $£3000$ , and give one reason the model may not be realistic.
[4 marks]
Answer $V = \dots\dots$ $\dots\dots$ years

Exam-style questions

Edexcel exam style

D1
A hedge grows at a constant rate. It is $1.35$ m tall $2$ weeks after it is planted and $1.83$ m tall after $8$ weeks. Priya says: “It grows $0.08$ m a week, so its height is directly proportional to the number of weeks.”
Priya is wrong.
(a)
Show that a model for the height $h$ m after $w$ weeks is $h = 0.08 w + 1.19$ .
[3 marks]
(b)
Explain why Priya is wrong.
[2 marks]
(c)
Hence find the least whole number of weeks after which the model gives a height over $2$ m.
[3 marks]
Answer $\dots\dots$ weeks
Total for D1: 8 marks

D2
The scatter graph shows the age $a$ years and the value $V$ thousand pounds of some cars of one model. A line of best fit passes through $P (2, 15.2)$ and $Q (7, 6.2)$ .
(a)
Use $P$ and $Q$ to find a model in the form $V = ma + c$ . Use it to estimate the value of a car that is $4.5$ years old.
[4 marks]
Answer $V = \dots\dots$ $£ \dots\dots$
(b)
Say what $- 1.8$ and $18.8$ mean.
[2 marks]
(c)
Hence explain why the model is not reliable for a car $12$ years old.
[2 marks]
Total for D2: 8 marks
D3
The price $£ P$ of a concert ticket and the number $Q$ of tickets are modelled by the demand line $P = 52 - 0.04 Q$ and the supply line $P = 0.02 Q + 7$ . The equilibrium point is where they meet.
(a)
Find the price and the number of tickets at the equilibrium point.
[3 marks]
Answer $Q = \dots\dots$ $P = \dots\dots$
(b)
A booking charge makes the supply line $P = 0.02 Q + 10$ . Hence find how many fewer tickets are sold.
[3 marks]
Answer $\dots\dots$ fewer
Total for D3: 6 marks
D4
Tank $A$ holds $2400$ litres and drains at $35$ litres a minute; tank $B$ holds $300$ litres and fills at $25$ litres a minute, from the same time. Model each volume after $t$ minutes, find when they are equal and that volume, and find the largest $t$ for which the model for $A$ can be used.
[6 marks]
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A phone company has three monthly plans. Plan A: $£12$ plus $5$ p a minute. Plan B: $£20$ plus $2$ p a minute. Plan C: $£35$ with no charge per minute. For how many different whole numbers of minutes is plan B the cheapest?
[6 marks]
Hint 1 · What to try
Write each cost in pence for $m$ minutes.
Hint 2 · The first line
B beats A once $m > 266.7$ ; B beats C while $m < 750$ .
Hint 3 · The full method
Count the whole numbers from $267$ to $749$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a straight-line model.
I can say what a and b mean.
I can test whether a linear model fits.
I can say when a model stops working.

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