OLEVEL
P2
A52 GRAPHICAL
SOLUTIONS WITH
MENSURATION
1
Answer the whole of this question on a sheet of graph paper.
r
A solid cylinder of radius r centimetres and height h centimetres has a
volume of 100 π cm3.
h
(a) (i) Show that h =
100
.
r2
[1]
(ii) The cylinder has a total surface area of πy square centimetres.
Show that y = 2r2 + 200 .
r
[1]
(b) The table below shows some values of r and the corresponding values of y, correct to the nearest
whole number.
r
1
1.5
2
3
4
5
6
y
202
138
108
85
82
90
p
(i) Find the value of p.
[1]
(ii) Using a scale of 2 cm to represent 1 cm, draw a horizontal r-axis for 1 r 6.
Using a scale of 2 cm to represent 20 cm2, draw a vertical y-axis for 70 y 220.
On your axes, plot the points given in the table and join them with a smooth curve.
[3]
(c) Use your graph to find the values of r for which y = 100.
[2]
(d) By drawing a tangent, find the gradient of the graph at the point where r = 2.
[2]
(e) Use your graph to find
(i) the value of r for which y is least,
[1]
(ii) the smallest possible value of the total surface area of the cylinder.
[1]
© UCLES 2004
4024/02/O/N/04
2
Answer the whole of this question on a sheet of graph paper.
x
The area of a rectangular garden, ABCD, is 100 m2.
A
Inside the garden there is a rectangular lawn, EFGH,
whose sides are parallel to those of the garden.
EF is 4m from AB.
FG, GH and HE are 1 m from BC, CD and DA respectively.
B
4
E
F
1
1
H
G
1
D
C
(a) Taking the length of AB to be x metres, write down expressions, in terms of x, for
(i) EF,
(ii) BC,
(iii) FG.
[2]
(b) Hence show that the area, y square metres, of the lawn, EFGH is given by
200
y = 110 – 5x – x .
[1]
(c) The table below shows some values of x and the corresponding values of y, correct to 1 decimal
place, where
y = 110 – 5x –
200
x .
x
4
5
6
7
8
9
10
y
p
45.0
46.7
46.4
45.0
42.8
40.0
Find the value of p.
[1]
(d) Using a scale of 2 cm to 1 metre, draw a horizontal x-axis for 4 < x < 10.
Using a scale of 2 cm to 2 square metres, draw a vertical y-axis for 40 < y < 48.
On your axes, plot the points given in the table and join them with a smooth curve.
[3]
(e) By drawing a tangent, find the gradient of the curve where x = 8.
[2]
(f) Use your graph to find
(i) the range of values of x for which the area of the lawn is at least 44 m2,
[2]
(ii) the value of x for which the area of the lawn is greatest.
[1]
© UCLES 2005
4024/02/O/N/05
3
Answer THE WHOLE of this question on a sheet of graph paper.
P
A
1
Q
S
1
D
x
10
B
C
1
R
The diagram represents a rectangular pond, ABCD, surrounded by a paved region.
The paved region has widths 1 m and 10 m as shown.
The pond and paved region form a rectangle PQRS.
The area of the pond is 168 m2.
(a) Taking the length of AB to be x metres, write down expressions, in terms of x, for
(i) PQ,
(ii) BC,
(iii) QR.
[2]
(b) Hence show that the area, y square metres, of the paved region, is given by
y = 22 + 11x + 336 .
x
[2]
(c) The table below shows some values of x and the corresponding values of y.
x
3
3.5
4
5
6
7
8
9
y
167
156.5
150
144.2
144
147
152
p
Calculate p.
[1]
(d) Using a scale of 2 cm to represent 1 metre, draw a horizontal x-axis for 3  x  9.
Using a scale of 2 cm to represent 5 square metres, draw a vertical y-axis for 140  y  170.
On your axes, plot the points given in the table and join them with a smooth curve.
(e) By drawing a tangent, find the gradient of the curve at (4, 150).
[3]
[2]
(f) Use your graph to find
(i) the smallest area of the paved region,
[1]
(ii) the length of PQ when the area of the paved region is smallest.
[1]
© UCLES 2009
4024/O2/M/J/09
4
Answer the whole of this question on a sheet of graph paper.
(a) The variables x and y are connected by the equation
y = 4x3 – 18x2 + 20x.
The table below shows some values of x and the corresponding values of y.
x
0
0.5
1
1.5
2
2.5
3
3.5
y
0
6
6
3
0
0
6
p
(i) Calculate the value of p.
[1]
(ii) Using a scale of 2 cm to represent 1 unit, draw a horizontal x-axis for 0 艋 x 艋 4.
Using a scale of 1 cm to represent 2 units, draw a vertical y-axis for – 4 艋 y 艋 24.
On your axes, plot the points given in the table and join them with a smooth curve.
[3]
(iii) Using your graph, find the values of x when y = 4.
[2]
(b) A rectangular card is 5 cm long and 4 cm wide.
As shown in the diagram, a square of side x centimetres is cut off from each corner.
5
x
x
4
The card is then folded to make an open box of height x centimetres.
(i) Write down expressions, in terms of x, for the length and width of the box.
[1]
(ii) Show that the volume, V cubic centimetres, of the box is given by the equation
V = 4x3 – 18x2 + 20x.
[2]
(iii) Which value of x found in (a)(iii) cannot be the height of a box with a volume of 4 cm3? [1]
(iv) Using the graph drawn in part (a)(ii), find
(a) the greatest possible volume of a box made from this card,
[1]
(b) the height of the box with the greatest volume.
[1]
© UCLES 2009
4024/02/O/N/09
5
x
A cuboid has a square cross-section, shown shaded in the diagram.
The length of the cuboid is x cm.
The sum of the length of the cuboid and one of the sides of the square is 10 cm.
(a) Show that the volume of the cuboid, y cm3, is given by y = x3 – 20x2 + 100x.
[2]
(b) The table shows some values of x and the corresponding values of y for
y = x3 – 20x2 + 100x.
x
1
2
3
4
5
6
y
81
128
147
144
125
96
7
8
9
9
(i) Complete the table.
[1]
(ii) On the grid opposite, plot the graph of y = x3 – 20x2 + 100x for 1 ⭐ x ⭐ 9.
[3]
(c) Use your graph to find
(i) the maximum volume of the cuboid,
Answer ................................. cm3 [1]
(ii) the possible values of x when the volume of the cuboid is 120 cm3.
Answer x = ............... or ...............[2]
y
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9 x
4
π r3]
3
For this part of the question take π as 3.
(d) [The volume of a sphere =
A sphere has a radius of 1 x cm.
2
By drawing a suitable graph on the grid, estimate the value of x when the sphere and the
cuboid have the same volume.
Answer x = ................................... [3]
4024/21/M/J/12
[The volume of a sphere = 43 π r3]
x
x
6
x
A solid consists of a sphere on top of a square-based cuboid.
The diameter of the sphere is x cm.
The base of the cuboid has sides of length x cm.
The sum of the height of the cuboid and one of the sides of the base is 8 cm.
(a) By considering the height of the cuboid, explain why it is not possible for this sphere to
have a radius of 5 cm.
Answer ......................................................................................................................................
............................................................................................................................................. [1]
(b) By taking the value of π as 3, show that the approximate volume, y cm3, of the solid is given by
y = 8x 2 –
x3
.
2
[2]
(c) The table below shows some values of x and the corresponding values of y for
x
1
2
y
7.5
28
y = 8x 2 –
x3
.
2
3
4
5
6
7
96
137.5
180
220.5
(i) Complete the table.
(ii) On the grid opposite, plot the graph of y = 8x 2 –
[1]
x3
for 1 艋 x 艋 7.
2
[3]
(iii) Use your graph to find the height of the cuboid when the volume of the solid is 120 cm3.
Answer ................................. cm [2]
y
240
220
200
180
160
140
120
100
80
60
40
20
0
0
1
2
3
4
5
6
7
x
(d) A cylinder has radius 3 cm and length x cm.
By drawing a suitable graph on the grid, estimate the value of x when the solid and the
cylinder have the same volume.
Take the value of π as 3.
Answer ........................................ [3]
© UCLES 2012
4024/22/M/J/12
7
Adil wants to fence off some land as an enclosure for his chickens.
The enclosure will be a rectangle with an area of 50 m2.
50 m2
x
(a) The enclosure is x m long.
Show that the total length of fencing, L m, required for the enclosure is given by
L = 2x +
100
.
x
[2]
(b) The table below shows some values of x and the corresponding values of L, correct to one
decimal place where appropriate, for L = 2x +
100
.
x
x
2
4
6
8
10
12
14
16
L
54
33
28.7
28.5
30
32.3
35.1
38.3
18
20
Complete the table.
[2]
(c) On the grid opposite
draw a horizontal x-axis for 0 G x G 20 using a scale of 1 cm to represent 2 m
and a vertical L-axis for 0 G L G 60 using a scale of 2 cm to represent 10 m.
On the grid, plot the points given in the table and join them with a smooth curve.
[3]
(d) Adil only has 40 m of fencing.
Use your graph to find the range of values of x that he can choose.
Answer
..................... G x G .................... [2]
(e) (i) Find the minimum length of fencing Adil could use for the enclosure.
Answer
................................................ m [1]
(ii) Find the length and width of the enclosure using this minimum length of fencing.
Give your answers correct to the nearest metre.
Answer Length = .................... m Width = .................... m [1]
(f) Suggest a suitable length and width for an enclosure of area 100 m2, that uses the minimum
possible length of fencing.
Answer Length = .................... m Width = .................... m [1]
© UCLES 2014
4024/22/M/J/14
8
[Curved surface area of a cone = πrl]
l
h
r
Thediagramshowsasolidconewithradiusrcm,heighthcmandslantheightlcm.
Sulemanmakessomesolidcones.
Theslantheightofeachofhisconesis4cmmorethanitsradius.
Use π = 3 throughout this question.
(a) Showthatthetotalsurfacearea,A cm2,ofeachofSuleman’sconesisgivenby A=6r(r+2).
[2]
(b) Completethetablefor A=6r(r+2).
r
0
1
A
0
18
2
3
4
5
6
144
210
288
[1]
(c) Onthegridopposite,drawthegraphof A=6r(r+2).
[2]
(d) Findanexpressionforhintermsofr.
Answer
h=...................................... [2]
A
300
250
200
150
100
50
0
1
2
3
4
5
6
r
(e) TheheightofoneofSuleman’sconesis12cm.
Calculateitsradius.
Answer
...................................... cm[2]
(f) AnotherofSuleman’sconeshasasurfaceareaof200cm2.
(i) Useyourgraphtofindtheradiusofthiscone.
Answer
...................................... cm[1]
(ii) Thisconeisplacedinaboxofheightpcm,wherepisaninteger.
Findthesmallestpossiblevalueofp.
p
Answer
© UCLES 2015
p=...................................... [2]
4024/21/O/N/15
9
3
x
The diagram shows the net of an open box of height 3 cm.
The area of the base of the box is 15 cm2.
The length of the rectangular base is x cm.
The total area of the net is A cm2.
(a) Show that A = 15 + 6x +
90
.
x
[2]
(b) Graham has one of these open boxes.
The total area of the net of his box is 65 cm2.
Write down an equation in x and solve it to find the length of the base of Graham’s box.
Give your answer correct to 2 decimal places.
Answer .................................... cm [4]
(c) (i) Complete the table below for A = 15 + 6x +
90
.
x
x
2
3
4
5
6
7
8
A
72
63
61.5
63
66
69.9
[1]
(ii) Draw the graph of A = 15 + 6x +
90
for 2 G x G 8 .
x
A
80
78
76
74
72
70
68
66
64
62
60
2
3
4
5
6
7
8 x
[2]
(iii) Delilah has one of these open boxes.
The area of the net of her box is 68 cm2.
Use your graph to find the length and width of Delilah’s box.
Answer length ......................... cm
width ......................... cm [2]
© UCLES 2018
4024/21/M/J/18
10 Zara fences off a piece of land next to a wall to make a vegetable garden.
Wall
Vegetable garden
x
The garden is a rectangle with the wall as one side of the rectangle.
The area of the garden is 18 square metres.
The width of the garden is x metres.
(a) The total length of fencing required for the garden is y metres.
Show that
y = 2x +
18
.
x
[1]
(b) (i) Complete the table for
x
y
1
2
y = 2x +
18
.
x
3
4
5
6
7
8
12
12.5
13.6
15
16.6
18.3
9
[2]
(ii) On the grid, draw the graph of y = 2x +
18
for 1 G x G 9 .
x
y
25
20
15
10
5
0
1
2
3
4
5
6
8
7
9
10 x
[3]
(c) Use your graph to find the two possible widths of the garden if 14 metres of fencing is used.
Answer ................. m or ................. m [2]
(d) The fencing costs $20 per metre.
(i) Find the minimum amount it will cost Zara to build the fence.
Answer $ ........................................... [2]
(ii) Zara wants to spend no more than $350 on the fence.
Find the greatest possible width of the garden Zara can make.
Answer �������������������������������������� m [2]
© UCLES 2018
4024/22/O/N/18
11
(8 – x)
30°
20
x
The diagram shows a triangular prism.
All lengths are in centimetres.
(a) Show that the volume, V cm3, of the prism is given by V = (40x - 5x 2) .
[3]
(b) On the grid on the next page, draw the graph of V = 40x - 5x 2 for 1 G x G 7 .
Three of the points have been plotted for you.
V
90
80
70
60
50
40
30
1
2
3
4
5
6
7
x
[3]
(c) Use your graph to find the possible values of x for one of these prisms with a volume of 50 cm3.
x = .................... or x = .................... [2]
(d) A cuboid has length 4 cm, width 3 cm and height x cm.
By drawing a suitable line on your graph, find the value of x when the prism and the cuboid have the
same volume.
x = ................................................... [3]
© UCLES 2019
4024/21/M/J/19