Question

1. the path of an acapulco cliff diver, as he dives into the sea, is given by the equation:

\\(h = -2d^2 + 2d + 36\\), where \\(h\\) metres is the divers height above the water, and \\(d\\) metres is the horizontal distance travelled by the diver.

a) determine the maximum height of the diver above sea level. (2 marks)

b) assume the cliff is vertical. how far from the base of the cliff will the diver enter the water? is it possible to dive this distance? (3 marks)

Explanation:

Find the horizontal distance at maximum height

Using the Parabolic Optimization knowledge point

$$ d = -\frac{b}{2a} = -\frac{2}{2(-2)} = 0.5\text{ m} $$

Calculate the maximum height

Using the Parabolic Optimization knowledge point

$$ h_{\text{max}} = -2(0.5)^2 + 2(0.5) + 36 = -0.5 + 1 + 36 = 36.5\text{ m} $$

Find the entry distance

Using the Quadratic Roots Application knowledge point

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Answer:

Question a

The maximum height of the diver above sea level is \(36.5\text{ metres}\).

Question b

The diver will enter the water at a horizontal distance of approximately \(4.77\text{ metres}\) from the base of the cliff.