Question

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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? how is it possible to dive this distance? (3 marks)

Explanation:

Model the parabolic arch

To solve Question 5, we model the parabolic arch on a Cartesian coordinate system. Let the ground be the \(x\)-axis and the line of symmetry of the parabola be the \(y\)-axis.

Since the arch is \(10\text{ m}\) wide at the base, the \(x\)-intercepts are at \(x = -5\) and \(x = 5\). The maximum height is \(4\text{ m}\), which occurs at the vertex \((0, 4)\).

The equation of the parabola is:

$$y = a(x - 5)(x + 5)$$

Using the vertex \((0, 4)\) to find \(a\):

$$4 = a(0 - 5)(0 + 5) \implies 4 = -25a \implies a = -0.16$$

Thus, the equation of the arch is:

$$y = -0.16x^2 + 4$$

Check tractor clearance

The trailer is \(2.5\text{ m}\) wide and passes through the center of the arch. This means the trailer extends from \(x = -1.25\) to \(x = 1.25\).

The critical points for clearance are at the outer edges of the trailer, \(x = \pm 1.25\). We calculate the height of the arch at these points:

$$y = -0.16(1.25)^2 + 4$$

$$y = -0.16(1.5625) + 4 = -0.25 + 4 = 3.75\text{ m}$$

Since the height of the hay bales is \(3.5\text{ m}\) and the arch height at the edge of the trailer is \(3.75\text{ m}\), we compare the values:

$$3.75\text{ m} > 3.5\text{ m}$$

The tractor can safely pass through.

Find the maximum height of the diver

For Question 6a, the path of the diver is given by:

$$h = -2d^2 + 2d + 36$$

This is a downward-opening parabola. The maximum height occurs at the vertex, where the horizontal distance is:

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

Substitute \(d = 0.5\) back into the equation to find the maximum height \(h\):

$$h = -2(0.5)^2 + 2(0.5) + 36$$

$$h = -2(0.25) + 1 + 36 = -0.5 + 37 = 36.5\text{ m}$$

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

Determine entry distance

For Question 6b, the diver enters the water when the height \(h = 0\):

$$-2d^2 + 2d + 36 = 0$$

Divide the entire equation by \(-2\):

$$d^2 - d - 18 = 0$$

Apply the quadratic formula:

$$d = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-18)}}{2(1)}$$

$$d = \frac{1 \pm \sqrt{1 + 72}}{2} = \frac{1 \pm \sqrt{73}}{2}$$

Since distance must be positive:

$$d = \frac{1 + \sqrt{73}}{2} \approx \frac{1 + 8.544}{2} \approx 4.77\text{ m}$$

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

Analyze physical feasibility

To answer how it is possible to dive this distance, we look at the horizontal velocity.

A horizontal distance of \(4.77\text{ m}\) is achieved because the diver runs and jumps outward from the cliff, giving them an initial horizontal velocity. Gravity acts vertically to pull them down, while their horizontal momentum carries them safely away from the vertical cliff face before entry.

Answer:

Question 5

Yes, the tractor can safely pass through. At the edge of the \(2.5\text{ m}\) wide trailer (\(1.25\text{ m}\) from the center), the arch height is \(3.75\text{ m}\), which is greater than the \(3.5\text{ m}\) height of the hay bales.

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. This distance is possible because the diver launches themselves horizontally off the cliff with an initial horizontal velocity, carrying them outward as they fall.