Chapter

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4 A friend is trying to calculate the volume of water in a reservoir amid fears there may be a severe water shortage. She comes up with the following expression:

[pic 84], where r is the amount of rain, d is how dry the area is, u is the usage of water by the townsfolk, and W is the volume of water in kL.

a Help her simplify the expression by simplifying each pronumeral one at a time.

b Does the final expression contain any potential surds?

c Express the fraction with a rational denominator.

d List the requirements for the possible values of r, u and d to give a rational answer.

e Calculate the volume of water in the reservoir when r = 4, d = 60 and u = 9. Write your answer in:

i [pic 85]

ii [pic 86]

iii [pic 87].

f Does a high value for d mean the area is dry? Explain using working.

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5 The speed of a toy plane can be modelled by the equation [pic 88], where

w = wind resistance and

p = battery power (from 0 (empty) to 10 (full)).

a Rationalise the denominator of the expression.

b Using your knowledge of perfect squares, estimate the speed of a toy plane with its battery half full and a wind resistance of 2. Check your answer with a calculator.

c How does the speed of the toy plane change with increasing wind resistance? Explain providing supportive calculations.

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Answers

Fluency

1 D

2 C

3 C

4 C

5 A

6 A

7 B

8 C

9 B

10 a [pic 89]

b [pic 90]

c [pic 91]

d [pic 92]

11 a 16

b [pic 93]

12 a [pic 94]

b [pic 95]

c [pic 96]

13 a 8

b [pic 97]

c 0

14 a [pic 98]

b [pic 99]

c [pic 100]

15 a 1

b 4

16 a [pic 101]

b [pic 102]

17 a [pic 103]

b [pic 104]

c [pic 105]

18 a 46

b [pic 106]

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Problem solving

1 [pic 107]

2 a 8

b 6

c 2

3 a [pic 108]

b The student has made two mistakes when squaring the left-hand bracket in line 1:32 = 9, 52 = 25.

4 a [pic 109]

b Yes, [pic 110]

c [pic 111]

d r should be a perfect square, u should be a perfect cube and d should be a rational number.

e i 0.0012346 kL

ii 1.2346 L

iii 1234.6 mL

f A high value for d causes the expression to be smaller, as d only appears on the denominator of the fraction. This means that when d is high there is less water in the reservoir and the area

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