A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 2: Quadratics · Lesson 4

Modelling with quadratics

71 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-modelling-with-quadratics/

Date

Do Now

Quick questions to start.

A1
Needed today
Write $4 x^{2} - 24 x + 29$ in the form $a (x + p)^{2} + q$ .
[2 marks]
Answer
A2
Needed today
Solve $5 x^{2} - 3 x - 4 = 0$ . Give your answers to 3 significant figures.
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A3
Last lesson
Find the range of values of $k$ for which $x^{2} - 10 x + k = 0$ has two distinct real roots.
[2 marks]
Answer $k \dots\dots$
A4
From chapter 1
Expand and simplify $(t - 4) (3 - 2 t)$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

A stone is thrown from the top of a cliff. Its height above the sea, $h$ m, after $t$ seconds is modelled by $h = 32 + 12 t - 5 t^{2}$ , $t \geq 0$ . Explain what the $32$ means, and find when the stone hits the sea.

Hint The

32

is the height when

t = 0

. The stone hits the sea when

h = 0

. Only a positive time makes sense.

B1
A ball is thrown from a window. Its height above the ground, $h$ m, after $t$ seconds is modelled by $h = 6.5 + 9 t - 4.9 t^{2}$ , $t \geq 0$ . Explain what the $6.5$ means, and find when the ball hits the ground, to 3 s.f.
[3 marks]
Answer $t = \dots\dots$

Example 2

The weekly cost, £ $C$ , of running a machine that makes $x$ hundred items is modelled by $C = a x^{2} + b x + c$ . When $x = 1$ , $C = 520$ ; when $x = 2$ , $C = 480$ ; when $x = 3$ , $C = 500$ . Find $a$ , $b$ and $c$ .

Hint Write three equations. Subtract them in pairs to remove

c

, then solve the two equations left.

B2
The depth, $d$ cm, of water in a tank $t$ minutes after a tap is opened is modelled by $d = a t^{2} + b t + c$ . When $t = 0$ , $d = 45$ ; when $t = 2$ , $d = 29$ ; when $t = 4$ , $d = 21$ . Find $a$ , $b$ and $c$ .
[3 marks]
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$

Example 3

A theatre sells $400$ tickets when each costs £ $20$ . For every £ $1$ off the price, it sells $25$ more tickets. The number sold at a price of £ $p$ is $n = M - 25 p$ . Find $M$ , and find the price that gives the greatest revenue.

Hint Put

p = 20

n = 400

to find

M

. Revenue is price

\times

number sold. Then complete the square.

B3
A café sells $150$ sandwiches a day at £ $4$ each. For every $10$ p off the price it sells $15$ more. The number sold at £ $p$ each is $n = M - 150 p$ . Find $M$ , the best price and the greatest revenue.
[4 marks]
Answer $M = \dots\dots$ £ $\dots\dots$ £ $\dots\dots$

Practice

A-Level

C1
A ball is thrown from a height of $1.44$ m. It reaches its greatest height, $7.2$ m, after $1.2$ seconds. Its height, $h$ m, after $t$ seconds is modelled by $h = A - B (t - C)^{2}$ . Find $A$ , $B$ and $C$ , and find when the ball lands on the ground, to 3 s.f.
[5 marks]
Answer $A = \dots\dots$ $B = \dots\dots$ $C = \dots\dots$ $t = \dots\dots$

C2
A ball's height, $h$ m, after $t$ seconds is modelled by $h = 1.2 + 7 t - 4.9 t^{2}$ . Use the discriminant to show that, according to the model, the ball never reaches a height of $4$ m.
[3 marks]
C3
The number of visitors to a museum in month $t$ after it opens is modelled by $V = 2400 + 220 t - 10 t^{2}$ . Find the value of $t$ at which the model predicts no visitors, and comment on using the model for later months.
[3 marks]
Answer $t = \dots\dots$

Exam-style questions

Edexcel exam style

D1
Priya writes: The ball hits the ground when $8 t - 5 t^{2} = 0$ , so $t (8 - 5 t) = 0$ and the ball is in the air for $1.6$ seconds.
A ball is thrown upwards from a balcony. Its height above the ground, $h$ m, after $t$ seconds is modelled by $h = 2.25 + 8 t - 5 t^{2}$ , $t \geq 0$ .
(a)
Explain the mistake Priya has made.
[2 marks]
(b)
Find when the ball hits the ground. Give your answer to 3 significant figures.
[3 marks]
Answer $t = \dots\dots$
(c)
Write $h$ in the form $A - B (t - C)^{2}$ . Hence find the greatest height of the ball, and for how long the ball is more than $4.25$ m above the ground, to 3 s.f.
[6 marks]
Answer $h = \dots\dots$ $\dots\dots$ m $\dots\dots$ s
Total for D1: 11 marks

D2
A farmer has $150$ m of fencing. She uses all of it to make a rectangular enclosure, split into two pens by one more fence, as in the diagram. Each short side is $x$ m and each long side is $y$ m.
Not drawn accurately
(a)
Show that the total area, $A$ m², of the enclosure is $A = 75 x - 1.5 x^{2}$ .
[2 marks]
(b)
Write $A$ in the form $q - p (x - r)^{2}$ . Hence find the greatest possible area, and the values of $x$ and $y$ that give it.
[5 marks]
Answer $A = \dots\dots$ greatest $\dots\dots$ m² $x = \dots\dots$ $y = \dots\dots$
(c)
Find the two values of $x$ for which the area is $864$ m².
[3 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
Total for D2: 10 marks
D3
A company's monthly profit, £ $P$ thousand, when it makes $x$ thousand items is modelled by $P = 36 x - 2 x^{2} - 70$ . Explain what the $- 70$ means. Find, in the form $a \pm b$ , the values of $x$ for which $P = 0$ . Hence find the least number of items the company must make to make a profit.
[5 marks]
Answer $x = \dots\dots$ $\dots\dots$ items
D4
A chain hangs between the tops of two posts, $26$ m and $14$ m high, standing $80$ m apart on level ground. Halfway between the posts the chain is $12$ m high. At $x$ m from the taller post the chain is $h$ m high, where $h = a x^{2} + b x + c$ .
(a)
Find the values of $a$ , $b$ and $c$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$
(b)
Hence find the least height of the chain above the ground, and how far it is from the taller post.
[4 marks]
Answer $\dots\dots$ m, $\dots\dots$ m from the taller post
(c)
A lamp hangs from the chain at the point, other than the top of the shorter post, where the chain is $14$ m high. How far is the lamp from the taller post?
[2 marks]
Answer $\dots\dots$ m
Total for D4: 10 marks

Extension

Stretch

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

E1
A ball is thrown straight up from the ground. Its height, $h$ m, after $t$ seconds is modelled by $h = u t - 5 t^{2}$ , where $u$ is a positive constant. The ball passes a window $20$ m above the ground twice, $3$ seconds apart. Find $u$ , and the greatest height of the ball.
[6 marks]
Hint 1 · What to try
Call the two times $t_{1}$ and $t_{2}$ . They are the roots of $5 t^{2} - u t + 20 = 0$ .
Hint 2 · The first line
$t_{1} + t_{2} = \frac{u}{5}$ and $t_{1} t_{2} = 4$ . Use $(t_{1} - t_{2})^{2} = (t_{1} + t_{2})^{2} - 4 t_{1} t_{2}$ .
Hint 3 · The full method
$\frac{u ^{2}}{25} - 16 = 9$ , so $u = 25$ . Then complete the square on $25 t - 5 t^{2}$ .
Answer $u = \dots\dots$ $\dots\dots$ m

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can explain a model's numbers.
I can find a model's greatest value.
I can fit a quadratic model to data.
I can say when a model fails.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers