Higher Mathematics
Optimisation
[SQA]
1. A goldsmith has built up a solid which consists of a triangular
prism of fixed volume with a regular tetrahedron at each end.
x
The surface area, A, of the solid is given by
√ 16
3 3
2
A( x ) =
x +
2
x
where x is the length of each edge of the tetrahedron.
Find the value of x which the goldsmith should use to
minimise the amount of gold plating required to cover the
solid.
Part
•1
•2
•3
•4
•5
•6
Marks
6
ss:
pd:
ss:
pd:
pd:
ic:
Level
A/B
Calc.
CN
Content
C11
•1
•2
•3
•4
•5
•6
know to differentiate
process
know to set f ′ ( x) = 0
deal with x−2
process
check for minimum
hsn.uk.net
Answer
x=2
Page 1
6
U1 OC3
2000 P2 Q6
A√′ ( x) = . . .
√
√ −2
3 3
−2
2 (2x − 16x ) or 3 3x − 24 3x
A′ ( x) = 0 √
− 16
or − 24x2 3
x2
x=2
x
2− 2 2+
′
A ( x) −ve 0 +ve
so x = 2 is min.
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Higher Mathematics
2. The parabolas with equations y = 10 − x2 and y = 52 (10 − x2 ) are shown in the
diagram below.
y
R
Q
S
T
P
x
O
y = 10 – x2
y = 25 (10 − x2 )
A rectangle PQRS is placed between the two parabolas as shown, so that:
• Q and R lie on the upper parabola.
• RQ and SP are parallel to the x-axis.
• T , the turning point of the lower parabola, lies on SP.
(a) (i) If TP = x units, find an expression for the length of PQ.
(ii) Hence show that the area, A, of rectangle PQRS is given by
A( x ) = 12x − 2x3 ·
3
(b) Find the maximum area of this rectangle.
Part
(ai)
(aii)
(b)
Marks
2
1
6
Level
B
B
C
Calc.
CN
CN
CN
Content
C11
C11
C11
6
Answer
6 − x2
2
2x × (6 −
√x ) = A( x)
max is 8 2
U1 OC3
2010 P2 Q5
•1 ss: know to and find OT
•2 ic: obtain an expression for PQ
•3 ic: complete area evaluation
•1 4
•2 10 − x2 − 4
•3 2x(6 − x2 ) = 12x − 2x3
•4
•5
•6
•7
•8
•9
•4
•5
•6
•7
•8
ss: know to and start to differentiate
pd: complete differentiation
ic: set derivative to zero
pd: obtain
ss: justify nature of stationary point
ic: interpret result and evaluate
area
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Page 2
A′ ( x) = 12 · · ·
12 − 6x2
2
12
√ − 6x = 0
2
√
x
···
2
A′ ( x) +
0
√
•9 Max and 8 2
···
−
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Higher Mathematics
[SQA]
3.
Part
(a)
(b)
(b)
Marks
2
3
3
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Level
C
C
A/B
Calc.
CN
CN
CN
Content
A6
C11
C11
Page 3
Answer
U1 OC3
1989 P2 Q7
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Higher Mathematics
[SQA]
4.
Part
(a)
(b)
(b)
Marks
4
3
5
Level
C
C
A/B
Calc.
NC
NC
NC
Content
CGD
C11
C11
Answer
U1 OC3
1994 P2 Q7
[END OF QUESTIONS]
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