Question

1. a country road passes through a parabolic arch used to support a train bridge. the arch is \\(10\text{ m}\\) wide at the base and the maximum height of the arch is \\(4\text{ m}\\). can a tractor pulling a trailer loaded with hay bales safely pass through the center of the arch if the trailer is \\(2.5\text{ m}\\) wide and height of the hay bales is \\(3.5\text{ m}\\) above the ground? (4 marks)

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:

Model the parabolic arch

Using the Quadratic Modeling knowledge point

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Evaluate tractor clearance

Using the Quadratic Word Problems knowledge point

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Find maximum height of the diver

Using the Parabolic Optimization knowledge point

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Calculate entry distance

Using the Quadratic Word Problems knowledge point

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Answer:

Question 5

Yes, the tractor pulling the trailer can safely pass. At the edge of the trailer (\(x = \pm 1.25\text{ m}\)), the height of the arch is \(3.75\text{ m}\), which is greater than the height of the hay bales (\(3.5\text{ m}\)).

Question 6

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

b) The diver will enter the water approximately \(4.77\text{ m}\) from the base of the cliff.