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 with two sides labeled as \\(3x\text{ cm}\\) and the base labeled as \\((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 m/s.

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

Explanation:

🆕 New Concept Discovered: Rearranging Literal Equations
Isolating a specific variable in a formula

Step 1: Isolate the term containing \( r \)

To make \( r \) the subject of the formula \( V = \pi r^2 h \), we first need to isolate the term \( r^2 \). We do this by dividing both sides of the equation by everything multiplied by \( r^2 \), which is \( \pi h \):

$$ \frac{V}{\pi h} = r^2 $$

Step 2: Solve for \( r \)

To isolate \( r \), we take the square root of both sides. Since the problem states that \( r \) is positive, we only consider the positive square root:

$$ r = \sqrt{\frac{V}{\pi h}} $$

Answer:

$$ r = \sqrt{\frac{V}{\pi h}} $$