Dora's Adventures II

PART II: THE ROYAL BRIDGE

Dora and Boots continue their journey along Hustleberry Lane until they notice a construction worker faced with a problem.

Construction Worker: Oh no. The Queen's Paddlewheel is journeying along this river, but the construction of the Royal Bridge isn't finished yet! I'm too tired and hungry to finish constructing the Royal Bridge, so I won't be able to finish constructing the bridge in time... What am I going to do?

Dora: That's not a problem! Boots and I can lend a helping hand, while you can go eat some of our delicious papayas.

Boots: Yes, well I think we have enough time to finish constructing the Royal Bridge before the Queen's Paddlewheel gets here. Give us the measurements, and we can finish the bridge in no time.

Construction Worker: Oh, thank you! The Royal Bridge must have a curved arch support that is 10 metres high and 16 metres wide. Well, I'm off then. Good day!

The construction worker took the papayas and walked down Hustleberry Lane.

Meanwhile...

Boots: Hey, Dora! We forgot to ask the construction worker what shape—a semi-circle, a semi-ellipse, a parabola, or a branch of a hyperbola—would give the best height available for the paddle steamer to pass safely beneath the bridge.

Dora: I guess the best height would have to be the shape that would give the maximum height possible, but which shape is it?

Dora and Boots are now faced with a problem. Help Dora and Boots solve the problem by answering the questions below.

QUESTIONS

a) The origin of the coordinate system has to be at the extreme left edge of the curved support. What is the equation of the curve in general form if the Royal Bridge is to be constructed in the shape of:

• a semi-circle?
• a semi-ellipse?
• a parabola?
• a branch of a hyperbola, where the length of the semi-transverse axis is 4 metres?

i. Semi-circle

Image 2.1

The equation of a circle in standard form is given, but only the top half of the circle will be drawn as part of the graph because the x-axis represents the ground. The bottom half of the bridge won't be built nor will the bridge be built underwater.

What We Know

• A point on the circle must be the origin of the coordinate system with the coordinates of (0, 0).
  
• Because the bridge has a maximum width of 16 metres, there is a point of intersection between the x-axis and the semi-circle with the coordinates (16, 0).
  
• The midpoint of these two coordinates is the centre of the circle with coordinates (8, 0).
  
• The radius should be 8 metres in length.
  
• Because the bridge has a maximum height of 10 metres, the coordinates of the maximum point on a circle is (8, 10).
  

Boots: Heeeey, wait a minute... These measurements don't make sense! If the radius of the circle along the x-axis is 8, then the distance between the centre of the circle to the maximum point on the circle must also be 8, not 10. Therefore, a semi-circle bridge with measurements of 10 metres as height and 16 metres as width is impossible.

You are right, Boots! I was just testing you.

ii. Semi-Ellipse

Image 2.2

The equation of a vertical ellipse in standard form is given, but only the top half of the ellipse will be drawn as part of the graph (making it a semi-ellipse) because the x-axis represents the ground. The bottom half of the bridge won't be built nor will the bridge be built underwater.

What We Know

• A point on the semi-ellipse must be the origin of the coordinate system with the coordinates of (0, 0).
  
• Because the bridge has a maximum width of 16 metres, there is a point of intersection between the x-axis and the semi-ellipse with the coordinates (16, 0).
  
• The midpoint of these two coordinates is the centre of the semi-ellipse with coordinates (8, 0).
  
• The semiminor axis, b, is 8 metres in length.
  
• Because the bridge has a maximum height of 10 metres, the coordinates of the maximum point on a circle is (8, 10).
• The semimajor axis, a, is 10 metres in length.
  
• The coordinates (x, y) represent any point on the semi-ellipse.
• The coordinates (h, k) represent the centre of the semi-ellipse.

Plug in these numbers into the equation, and an equation for the semi-ellipse is made. See Image 2.2.

iii. Parabola

Image 2.3

The equation of a vertical parabola in standard form is given, but only the parabola that is 10 metres above the ground will be drawn as part of the graph because the x-axis represents the ground. The bridge will not extend underwater nor will the bridge extend underground.

What We Know

• A point on the parabola must be the origin of the coordinate system with the coordinates of (0, 0). This is one of the parabola's roots.
  
• Because the bridge has a maximum width of 16 metres, there is a point of intersection between the x-axis and the parabola with the coordinates (16, 0). This is one of the parabola's roots.
  
• The midpoint of these two coordinates is found along the axis of symmetry (has the same x-coordinate as the vertex of the parabola) with coordinates (8, 0).
  
• The distance between the midpoint of the two ends of the bridge to one of either root is 8 metres in length.
  
• Because the bridge has a maximum height of 10 metres, the coordinates of the vertex is (8, 10).
• The distance between the midpoint of the two roots to the vertex is 10 metres in length.
• The coordinates (x, y) represent any point on the parabola.
• The coordinates (h, k) represent the vertex of the parabola.

Plug in these numbers into the equation, determine the distance between the focal point to the vertex by solving for p, and an equation for the parabola is made. See Image 2.3.

iv. Hyperbola

Image 2.4

The equation of a vertical hyperbola in standard form is given, but only the lower branch of the hyperbola that is 10 metres above the ground will be drawn as part of the graph because the x-axis represents the ground. The bridge will not extend underwater nor will the bridge extend underground nor will the bridge open up and hang suspended in mid-air.

What We Know

• A point on the hyperbola must be the origin of the coordinate system with the coordinates of (0, 0).
  
• Because the bridge has a maximum width of 16 metres, there is a point of intersection between the x-axis and the hyperbola with the coordinates (16, 0).
  
• The midpoint of these two coordinates is has coordinates (8, 0).
  
• The distance between the midpoint of the two ends of the bridge to one of either end of the bridge is 8 metres in length.
  
• Because the bridge has a maximum height of 10 metres, the coordinates of the vertex is (8, 10).
• The distance between the midpoint of the two ends of the bridge to the vertex is 10 metres in length.
• Because the length of the semitransverse axis, a, is 4 metres, the coordinates of the centre of the hyperbola is (8, 14).
  
• The coordinates (x, y) represent any point on the hyperbola.
• The coordinates (h, k) represent the centre of the hyperbola.

Plug in these numbers into the equation, determine the length of the semiconjugate axis by solving for b, and an equation for the hyperbola is made. See Image 2.3.

b) The Queen's Paddlewheel is a paddle steamer owned by the queen. She boards onto the paddle steamer on her voyage to Prince Livingston's Castle located near the other end of the river where the Royal Bridge is being constructed. The paddle steamer is measured to be 8 metres high above the water and 10 metres wide. Which of the four segments of the conic shapes—a semi-circle, a semi-ellipse, a parabola, a branch of a hyperbola—in a) would allow the paddle steamer to pass safely beneath the bridge, thus ensuring the safety of the queen?

Image 2.5

The width of the Royal Bridge is 16 metres while the height of the Royal Bridge is 10 metres.

The width of the Queen's Paddlewheel is 8 metres for height and 10 metres for width.

Dora and Boots wants to know which structure would give enough space for the paddle steamer to travel beneath the bridge. Because the width, x, of the paddle steamer is 10 metres, the height, y, of the paddle steamer must not be 8+ metres.

Plug in the numbers to where x is in the equation to get y. Reject all negative y-values because the boat can't have a negative height; the boat isn't underwater.

If the bridge were to have an elliptical shape, the height of the paddle steam can have a height up to sqrt(375/4) = 9.68 metres. The clearance is 1.68 metres (9.68 - 8 = 1.68). Thus, the paddle steamer won't collide into the bridge, and the queen is safe.

If the Royal Bridge were to have a parabolic shape, the height of the Queen's Paddlewheel can have a height up to 8 metres; the clearance is 0 metres (8 - 8 = 0). Thus, the paddle steamer will collide into the bridge, and the queen may drown.

If the Royal Bridge were to have a hyperbolic shape, the height of the paddle steam can have a height up to 8 metres; the clearance is 0 metres (8 - 8 = 0). Thus, the paddle steamer will collide into the bridge, and the queen may drown.

In conclusion, the best choice for the shape of the bridge is to construct the bridge in the shape of a semi-ellipse.