Linear function from conditions

5 The function f is defined by \(\mathrm { f } : x \mapsto a x + b\), for \(x \in \mathbb { R }\), where \(a\) and \(b\) are constants. It is given that \(f ( 2 ) = 1\) and \(f ( 5 ) = 7\).

1. Find the values of \(a\) and \(b\).
2. Solve the equation \(\operatorname { ff } ( x ) = 0\).

2. A tree was planted in the ground. Exactly 2 years after it was planted, the height of the tree was 1.85 m . Exactly 7 years after it was planted, the height of the tree was 3.45 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted in the ground, can be modelled by the equation $$H = a t + b$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\).
2. State, according to the model, the height of the tree when it was planted.
   

Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]

Make \(b\) the subject of the following formula. $$a = \frac{3}{5}b^2c$$ [3]

Rearrange the following formula to make \(r\) the subject, where \(r > 0\). $$V = \frac{1}{3}\pi r^2(a + b)$$ [3]