The Robert Smyth School
Mathematics Faculty
Topic 11
Direct Proportionality
Innovation & excellence
Homework on Direct Proportion - Higher
1.
A company sells circular badges of different sizes. The price, P pence, of a
badge is proportional to the square of its radius, r cm. The price of a badge of
radius 3 cm is 180 pence.
(a)
Find an equation expressing P in terms of r.
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Answer ........................................
(3)
(b)
Calculate the price of a badge of radius 4 cm.
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Answer ........................................ pence
(1)
2.
y is proportional to x where x > 0 and y > 0.
When x = 5, y = 12.5.
(a)
Find an equation expressing y in terms of x.
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Answer y = ..............................................
(3)
(b)
Calculate x when y = 0.72.
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Answer x = ..............................................
(2)
3.
W and P are both positive quantities.
W is directly proportional to the square root of P.
When W = 12, P = 16.
(a)
Express W in terms of P.
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The Robert Smyth School
Mathematics Faculty
Topic 11
Direct Proportionality
Innovation & excellence
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Answer ..........................................................
(3)
(b)
What is the value of W when P = 25?
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Answer ..........................................................
(1)
(c)
What is the value of P when W = 21?
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Answer ..........................................................
(2)
4.
In an experiment measurements of t and h were taken.
These are the results.
t
2
5
6
h
10
62.5
90
Which of these rules fits the results?
(A) h α t
(B) h α t
2
(C) h α t
3
You must show all your working.
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Answer ............................................
(Total 4 marks)
5.
Cheese is sold in different sizes. The weight of each cheese, W kilograms, is
proportional to the cube of its height, h centimetres. A cheese of weight 10 kg
has a height of 12 cm.
The Robert Smyth School
Mathematics Faculty
Topic 11
Direct Proportionality
Innovation & excellence
(a)
Find an equation connecting W and h.
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Answer ........................................................................................
(3)
(b)
Another cheese has a height of 6 cm. Find its weight.
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Answer ................................................................................ kg (1)
(c)
Find the height of a cheese that has a weight of 20 kg.
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Answer ................................................................................... cm
(3)
(Total 7 marks)
6.
The area, A square metres, of a new logo is directly proportional to the square
of its width, w metres.
The area of the logo is 12 square metres when its width is 4 metres.
(a)
Find an equation connecting A and w.
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Answer ..........................................................................
(3)
(b)
Find the area of a logo with a width of 5 metres.
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Answer ................................................................... m
2
(1)
The Robert Smyth School
Mathematics Faculty
Topic 11
Direct Proportionality
Innovation & excellence
7.
Salt is sold in different sized blocks.
The weight of each block, B kilograms, is directly proportional to the cube of
its height, h metres.
A block of weight 54 kg has height 3m.
(a)
Find an equation connecting h and B.
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Answer
(b)
B  ..................................................
..........
(3)
Find the weight of a block with a height of 1m.
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Answer ................................................ kg
(1)
(c)
Another block has a weight of 128 kg.
Find its height.
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Answer ................................................
(3)
Answers
1
(a)
P = kr
2
o.e. M1
2
e.g. P a r
If (a) blank, mark (a) within (b) for M marks
only
2
180 = k × 3
o.e. M1
The Robert Smyth School
Mathematics Faculty
Topic 11
Direct Proportionality
Innovation & excellence
2
or 20 seen or 20r seen
P = 20r
(b)
2
o.e. A1
320
B1 ft
Allow £3.20
k
2
ft only if kr used or 2 in (a) and
k
r
1
2
If P = kr in (a) only and 320 as answer in (b) –
nd
award 2 M mark in (a)
[4]
2
(a)
y = kx
2
o.e.M1
12.5
= k or 12.5 = k × 25
25
o.e.M1
or k = 0.5
y = 0.5x
2
o.e. A1
or y 0.5x
(b)
2
o.e. M2
o.e. M1 ft
0.72 / 0.5
or (x =) their rearrangement and substitution
1.2
3
(a)
W
A1 cao
[5]
P or W = k P M1
12
16 or 12 = k 16 acceptable
for M1
k=3
A1
W=3 P
A1
ft their k, but must be formally stated
2
Accept equivalent form eg. P = (W/3)
(b)
W = 15
B1
ft. their k
(c)
P = 21 ÷ 3 or P = 7
allow 21 ÷ (their k)
B1
P = 49 B1
ft. for their P value “ squared”
[6]
4
2
Test the data using h = kt, h = kt
3
or h = kt M1
must find a value of ‘k’
Test one of the laws to reach a correct conclusion
2
If h = kt the first one tested (correctly) then
award A2
A1
Test a second law to reach a correct conclusion
A1
Select h = kt
2
A1
The Robert Smyth School
Mathematics Faculty
Topic 11
Direct Proportionality
Innovation & excellence
Conclusion must be stated
No working ... no marks
[4]
5
(a)
3
3
h or W = kh M1
W
3
10 = 12 k
k=
10
1728
W=
M1
10
3
3
× h or W = 0.005 787 h
1728
A1
B1
Accept 0.00579
(b)
W=
10
3
× 6 = 1.25
1728
(c)
20 =
10 3
h
1728
3
h = 3456
M1
h = 3 3456
= 15.1...
M1
A1
Accept 12 3 2
[7]
6
(a)
(b)
2
2
A w or A = Kw M1
When A = 12, w = 4,
12 = 16K
K=
3
4
M1
A=
3 2
w
4
A1
When w = 5, A =
= 18.75 m
2
3 2
.5
4
B1
(ft if first M1 gained)
[4]
7
(a)
(b)
3
3
B h or B = kh
M1
When B = 54, h = 3
3
54 = 27k or 3 k M1
k=2
3
\ B = 2h
A1
When h = 1, B = 2 × 1 = 2
3
From h
B1 ft
The Robert Smyth School
Mathematics Faculty
Topic 11
Direct Proportionality
Innovation & excellence
(c)
When B = 128,
3
128 = 2h
From h
M1 ft
3
3
h = 64
M1 ft
3
From h
h = 4 metres
A1
Must have units
[7]