Question

part b
18 free-response questions
60 marks
show your working where appropriate.

21 (3 marks) if \\(v = \pi r^2 h\\) and \\(r\\) is positive, change the subject of the formula to \\(r\\).

22 (2 marks) solve \\(21 - 3p = 6\\).

23 (3 marks) solve \\(3y^2 = 108\\).

24 (4 marks) solve \\(5(w - 8) = 2(w + 4)\\).

25 (3 marks) solve \\(\frac{2p}{5} + 8 = 10\\).

26 (4 marks) write each of the following statements algebraically as an equation (but do not solve).
a if \\(d\\) is multiplied by 4, and 5 is added, the answer is 17.
b 3 less than \\(y\\) is equal to 7.

27 (4 marks) solve \\(\frac{3p}{2} - \frac{p}{3} = 4\\).

28 (3 marks) the sum of the angles (in degrees) of a polygon with \\(n\\) sides is given by the formula \\(a = 180(n - 2)\\).
how many sides has a polygon with an angle sum of \\(1080^\circ\\)?

29 (3 marks) the perimeter of this isosceles triangle is 57 cm. find the value of \\(x\\).
an isosceles triangle is shown with two equal sides labeled \\(3x\text{ cm}\\) and the base labeled \\((2x + 1)\text{ cm}\\).

30 (2 marks) the braking distance, \\(d\text{ m}\\), of a bicycle travelling at speed \\(v\text{ m/s}\\) is:
\\d = \frac{v(v + 1)}{2}\\
find the braking distance at a speed of \\(6.2\text{ m/s}\\).

31 (4 marks) the sum of three consecutive numbers is 51. use an equation to find the three numbers.

Explanation:

⚡ Using what you learned: understanding linear equations

Step 1: Solve Question 21

Rearrange \( V = \pi r^2 h \) to make \( r \) the subject, given \( r > 0 \):
\[ \frac{V}{\pi h} = r^2 \]
\[ r = \sqrt{\frac{V}{\pi h}} \]

Step 2: Solve Question 22

Solve \( 21 - 3p = 6 \):
\[ -3p = 6 - 21 \]
\[ -3p = -15 \]
\[ p = 5 \]

Step 3: Solve Question 23

Solve \( 3y^2 = 108 \):
\[ y^2 = 36 \]
\[ y = \pm 6 \]

Step 4: Solve Question 24

Solve \( 5(w - 8) = 2(w + 4) \):
\[ 5w - 40 = 2w + 8 \]
\[ 3w = 48 \]
\[ w = 16 \]

Step 5: Solve Question 25

Solve \( \frac{2p}{5} + 8 = 10 \):
\[ \frac{2p}{5} = 2 \]
\[ 2p = 10 \]
\[ p = 5 \]

Step 6: Solve Question 26

Translate statements into algebraic equations:

• a "If \( d \) is multiplied by 4, and 5 is added, the answer is 17":

\[ 4d + 5 = 17 \]

• b "3 less than \( y \) is equal to 7":

\[ y - 3 = 7 \]

Step 7: Solve Question 27

Solve \( \frac{3p}{2} - \frac{p}{3} = 4 \). Multiply all terms by the lowest common denominator, 6:
\[ 3(3p) - 2(p) = 24 \]
\[ 9p - 2p = 24 \]
\[ 7p = 24 \]
\[ p = \frac{24}{7} \]

Step 8: Solve Question 28

Solve \( 180(n - 2) = 1080 \):
\[ n - 2 = 6 \]
\[ n = 8 \]

Step 9: Solve Question 29

The triangle is isosceles. Based on the markings, the two sides of length \( 3x\text{ cm} \) are equal. The perimeter is \( 57\text{ cm} \):
\[ 3x + 3x + (2x + 1) = 57 \]
\[ 8x + 1 = 57 \]
\[ 8x = 56 \]
\[ x = 7 \]

Step 10: Solve Question 30

Substitute \( V = 6.2 \) into \( d = \frac{V(V + 1)}{2} \):
\[ d = \frac{6.2(6.2 + 1)}{2} \]
\[ d = \frac{6.2 \times 7.2}{2} \]
\[ d = 3.1 \times 7.2 = 22.32 \]

Step 11: Solve Question 31

Let the three consecutive numbers be \( x \), \( x + 1 \), and \( x + 2 \):
\[ x + (x + 1) + (x + 2) = 51 \]
\[ 3x + 3 = 51 \]
\[ 3x = 48 \]
\[ x = 16 \]
The three numbers are \( 16 \), \( 17 \), and \( 18 \).

Answer:

• 21: \( r = \sqrt{\frac{V}{\pi h}} \)
• 22: \( p = 5 \)
• 23: \( y = \pm 6 \)
• 24: \( w = 16 \)
• 25: \( p = 5 \)
• 26 a: \( 4d + 5 = 17 \)
• 26 b: \( y - 3 = 7 \)
• 27: \( p = \frac{24}{7} \) (or \( 3\frac{3}{7} \))
• 28: \( 8 \) sides
• 29: \( x = 7 \)
• 30: \( 22.32\text{ m} \)
• 31: \( 16, 17, 18 \)