11. An archer shoots an arrow.
The height, \(H\) metres, of the arrow above the ground is modelled by the formula
$$H = 1.8 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$
where \(d\) is the horizontal distance of the arrow from the archer, measured in metres.
Given that the arrow travels in a vertical plane until it hits the ground,
- find the horizontal distance travelled by the arrow, as given by this model.
- With reference to the model, interpret the significance of the constant 1.8 in the formula.
- Write \(1.8 + 0.4 d - 0.002 d ^ { 2 }\) in the form
$$A - B ( d - C ) ^ { 2 }$$
where \(A , B\) and \(C\) are constants to be found.
It is decided that the model should be adapted for a different archer.
The adapted formula for this archer is
$$H = 2.1 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$
Hence or otherwise, find, for the adapted model - the maximum height of the arrow above the ground.
- the horizontal distance, from the archer, of the arrow when it is at its maximum height.