A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 12: Differentiation · Lesson 6

Modelling with differentiation

72 marks

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

Date

Do Now

Quick questions to start.

A1
Needed today
A cuboid has a square base of side $x$ cm and a volume of $96$ cm³. Show that its surface area is $2 x^{2} + \frac{384}{x}$ cm².
[2 marks]
Answer
A2
Last lesson
Find the coordinates of the stationary point of $y = 2 x + \frac{18}{x}$ , $x > 0$ .
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A3
From chapter 2
Find the greatest value of $40 t - 5 t^{2}$ , and the value of $t$ at which it happens.
[2 marks]
Answer $\dots\dots$ when $t = \dots\dots$
A4
From chapter 1
Write $\frac{3 x ^{2} - 8}{2 x}$ in the form $a x^{p} + b x^{q}$ .
[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 reservoir holds $V = 50 + 6 t^{2} - \frac{1}{3} t^{3}$ million litres $t$ weeks into a wet season, $0 \leq t \leq 15$ . Find $\frac{d V}{d t}$ at $t = 2$ and say what it means. When is $V$ rising fastest, and how fast?

Hint The rate

\frac{d V}{d t}

is greatest where its own derivative is zero.

B1
A culture holds $N = 400 + 30 t^{2} - t^{3}$ bacteria $t$ hours after the start, $0 \leq t \leq 25$ . Find the rate of increase of $N$ at $t = 4$ , with its units. When is $N$ increasing fastest, and how fast?
[5 marks]

Example 2

A closed box has a square base of side $x$ cm and surface area $216$ cm². Show that its volume is $V = 54 x - \frac{1}{2} x^{3}$ cm³. Find the greatest volume; show it is a maximum.

Hint Write the height in terms of

x

first.

B2
An open box has a square base of side $x$ cm and volume $32$ cm³. Show that its area is $A = x^{2} + \frac{128}{x}$ cm². Find the least area; show it is a minimum.
[5 marks]

Example 3

A closed tin is a cylinder of radius $r$ cm and volume $250 π$ cm³. Show that its surface area is $S = 2 π r^{2} + \frac{500 π}{r}$ cm². Find the exact least surface area, and show that it is a minimum.

Hint Write the height in terms of

r

. Keep

π

π

B3
A cylinder with a base but no lid has radius $r$ cm and surface area $75 π$ cm². Show that its volume is $V = \frac{π}{2} (75 r - r^{3})$ cm³. Find the exact greatest volume, and show that it is a maximum.
[5 marks]

Practice

A-Level

C1
Water is poured into a conical funnel. When the depth of water is $h$ cm, the volume of water is $V = \frac{π}{12} h^{3}$ cm³. Find $\frac{d V}{d h}$ when $h = 6$ , and explain what your answer means.
[2 marks]
Answer $\frac{d V}{d h} = \dots\dots$

C2
A lorry makes a journey of $240$ km at a steady speed of $v$ km/h. The driver is paid £ $20$ per hour, and the running costs are £ $\frac{v}{80}$ for each kilometre. Show that the total cost of the journey, £ $C$ , is $C = \frac{4800}{v} + 3 v$ . Find the speed that makes the cost least and the least cost, and show that it is a minimum.
[5 marks]
C3
The point $P (x, y)$ lies on the curve $y = 3 - \frac{1}{4} x^{2}$ , and $O$ is the origin. Show that $O P^{2} = \frac{1}{16} x^{4} - \frac{1}{2} x^{2} + 9$ . Hence find the shortest distance from $O$ to the curve.
[5 marks]
Not drawn accurately
Answershortest distance $= \dots\dots$

Exam-style questions

Edexcel exam style

D1
A window is a rectangle with a semicircle on top, as shown. The rectangle is $2 r$ m wide and $y$ m tall, and the semicircle has radius $r$ m. The perimeter of the whole window is $10$ m.
Not drawn accurately
(a)
Show that the area of the window, $A$ m², is $A = 10 r - 2 r^{2} - \frac{1}{2} π r^{2}$ .
[3 marks]
(b)
Use calculus to find the exact value of $r$ for which $A$ is greatest, and show that it is a maximum.
[5 marks]
Answer $r = \dots\dots$
Total for D1: 8 marks

D2
A box with no lid is made from a rectangular sheet of card $48$ cm by $30$ cm, by cutting a square of side $x$ cm from each corner and folding up the sides. Sam finds that $\frac{d V}{d x} = 0$ when $x = 6$ and when $x = 20$ , and writes: "The box is biggest when $x = 20$ , because $20$ is the larger value."
Sam's conclusion is wrong.
Not drawn accurately
(a)
Show that the volume of the box, $V$ cm³, is $V = 4 x^{3} - 156 x^{2} + 1440 x$ .
[2 marks]
(b)
Explain why $x = 20$ is not possible. Find the greatest volume of the box, and show that it is a maximum.
[6 marks]
Answergreatest volume $= \dots\dots$ cm³
Total for D2: 8 marks
D3
A closed box is a cuboid with a base $x$ cm by $2 x$ cm and a volume of $576$ cm³. Its total surface area is $S = 4 x^{2} + \frac{1728}{x}$ cm². Find the least surface area, and show that it is a minimum.
[5 marks]
Answerleast surface area $= \dots\dots$ cm²
D4
The number of fish, $P$ , in a lake $t$ weeks after it was stocked is modelled by $P = 500 + 300 t - 15 t$ , $t \geq 0$ .
(a)
Find $\frac{d P}{d t}$ , and the rate at which the number of fish is changing when $t = 25$ . Say what this rate means.
[4 marks]
Answer $\dots\dots$ fish per week
(b)
Find the greatest number of fish that the model predicts, and show that it is a maximum.
[4 marks]
Answergreatest number $= \dots\dots$
(c)
Find when the model predicts $P = 500$ again. Give one reason why the model may not be suitable after that.
[2 marks]
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 cone has its vertex and the whole circle of its base on a sphere of radius $R$ . Find the greatest possible volume of the cone as a fraction of the volume of the sphere.
[6 marks]
Hint 1 · What to try
Let the cone have height $h$ and base radius $r$ . Use the centre of the sphere to find $r^{2}$ .
Hint 2 · The first line
The centre is $h - R$ above the base, so $r^{2} = R^{2} - (h - R)^{2} = 2 R h - h^{2}$ , and the volume is $\frac{π}{3} (2 R h^{2} - h^{3})$ .
Hint 3 · The full method
$\frac{d V}{d h} = \frac{π}{3} (4 R h - 3 h^{2}) = 0$ gives $h = \frac{4 R}{3}$ . Then $V = \frac{32 π R ^{3}}{81}$ ; divide by $\frac{4}{3} π R^{3}$ .
Answerfraction $= \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find a rate of change, say what it means, and find when it is greatest.
I can use a fixed area or volume to write a quantity in one variable, and make it greatest or least.
I can show a value is a maximum or a minimum, and check it makes sense.

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