(a)

\begin{align} y & = -0.2x^2 + 6x + 1.4 \\ & = -0.2(x^2 - 30x) + 1.4 \\ & = -0.2\left[x^2 - 30x + \left(30 \over 2\right)^2 - \left(30 \over 2\right)^2 \right] + 1.4 \\ & = -0.2( x^2 - 30x + 15^2 - 15^2 ) + 1.4 \\ & = -0.2 [ (x - 15)^2 - 225 ] + 1.4 \\ & = -0.2 (x - 15)^2 + 45 + 1.4 \\ & = -0.2 (x - 15)^2 + 46.4 \\ \\ [ & \text{Max. point: } (15, 46.4) ] \\ \\ \text{Greatest height } (y) & = 46.4 \text{ m} \\ \\ \text{Corresponding distance } (x) & = 15 \text{ m} \end{align}

(b)

\begin{align} y & = -0.2 (x - 15)^2 + 46.4 \\ \\ \text{Let } & y = 1.4, \\ 1.4 & = -0.2(x - 15)^2 + 46.4 \\ 0.2(x - 15)^2 & = 46.4 - 1.4 \\ 0.2(x - 15)^2 & = 45 \\ (x - 15)^2 & = {45 \over 0.2} \\ (x - 15)^2 & = 225 \\ x - 15 & = \pm \sqrt{225} \\ x - 15 & = \pm 15 \\ \\ x & = 15 + 15 \phantom{0} \text{ or } \phantom{0} -15 + 15 \\ x & = 30 \phantom{0} \text{ or } \phantom{0} 0 \end{align}

$$ \therefore \text{Horizontal distance travelled} = 30 \text{ m} $$

(a)

\begin{align} y & = - 4x^2 + 18x + c \\ & = - 4 \left( x^2 - {9 \over 2}x \right) + c \\ & = - 4 \left[ x^2 - {9 \over 2}x + \left(9 \over 4\right)^2 - \left(9 \over 4\right)^2 \right] + c \\ & = - 4 \left[ \left(x - {9 \over 4}\right)^2 - {81 \over 16} \right] + c \\ & = - 4 \left(x - {9 \over 4}\right)^2 + {81 \over 4} + c \\ \\ \text{Maximum value} & = {81 \over 4} + c, \text{ when } x = {9 \over 4} \\ \\ {81 \over 4} + c & = 30 \\ c & = 30 - {81 \over 4} \\ c & = {39 \over 4} \end{align}

(b)

\begin{align} y & = - 4 \left(x - {9 \over 4}\right)^2 + {81 \over 4} + c \\ & = - 4 \left(x - {9 \over 4}\right)^2 + {81 \over 4} + {39 \over 4} \\ & = - 4 \left(x - {9 \over 4}\right)^2 + 30 \\ \\ \text{Let } & x = 3, \\ y & = -4 \left(3 - {9 \over 4}\right)^2 + 30 \\ & = {111 \over 4} \\ \\ \text{Height of ball} & = {111 \over 4} \text{ m} \end{align}

$$ \therefore \text{Ball is moving downwards} $$

(a)

\begin{align} \text{Perimeter} & = 400 \text{ m} \\ \\ \text{Breadth} & = {400 - 3x - 3x \over 2} \\ & = {400 - 6x \over 2} \\ & = {2(200 - 3x) \over 2} \\ & = (200 - 3x) \text{ m} \\ \\ \text{Area} & = 3x \times (200 - 3x) \\ & = (600x - 9x^2) \text{ m}^2 \end{align}

(b)

\begin{align} \text{Area} & = 600x - 9x^2 \\ & = -9x^2 + 600x \\ & = -9 \left(x^2 - {200 \over 3}x\right) \\ & = -9 \left[ x^2 - {200 \over 3}x + \left(100 \over 3\right)^2 - \left(100 \over 3\right)^2 \right] \\ & = -9 \left[ \left(x - {100 \over 3}\right)^2 - {10 \phantom{.}000 \over 9} \right] \\ & = -9 \left(x - {100 \over 3}\right)^2 + 10 \phantom{.} 000 \\ \\ \text{Max. area} & = 10 \phantom{.}000 \text{ m}^2 \text{ when } x = {100 \over 3} = 33{1 \over 3} \\ \\ \therefore \text{Area } & \text{cannot exceed } 10 \phantom{.} 000 \text{ m}^2 \end{align}

(i)

\begin{align} y & = {5 \over 4}x^2 - 20x + 340 \\ \\ \text{When } & x = 0, \\ y & = {5 \over 4}(0)^2 - 20(0) + 340 \\ & = 340 \\ \\ 340 \text{ is } & \text{the fixed cost of assembly} \end{align}

(ii)

\begin{align} y & = {5 \over 4}x^2 - 20x + 340 \\ \\ \text{When } & x = 7, \\ y & = {5 \over 4}(7)^2 - 20(7) + 340 \\ & = \$ 261.25 \\ \\ \text{When } & x = 9, \\ y & = {5 \over 4}(9)^2 - 20(9) + 340 \\ & = \$ 261.25 \\ \\ \\ x & = {7 + 9 \over 2} \\ & = 8 \\ \\ x = 8 \text{ is the } & \text{line of symmetry to the graph of } y = {5 \over 4}x^2 - 20x + 340 \end{align}

(iii)

\begin{align} y & = {5 \over 4}x^2 - 20x + 340 \\ & = {5 \over 4}(x^2 - 16x) + 340 \\ & = {5 \over 4} \left[ x^2 - 16x + \left(16 \over 2\right)^2 - \left(16 \over 2\right)^2 \right] + 340 \\ & = {5 \over 4} ( x^2 - 16x + 8^2 - 8^2 ) + 340 \\ & = {5 \over 4} [ (x - 8)^2 - 64 ] + 340 \\ & = {5 \over 4} (x - 8)^2 - 80 +34 0\\ & = {5 \over 4} (x - 8)^2 + 260 \\ \\ \text{Minimum cost } (y) & = 260 \text{ when } x = 8 \\ \\ \therefore \text{Not possible } & \text{for cost to be } \$ 250 \end{align}