Fit an elliptical tunnel from boundary points

The diagram below shows the cross-section of a single track tunnel entrance for steam-engine in a model train system. The elliptical entrance of the tunnel can be modelled by the equation x2+Ay2+By+C=0x^2+Ay^2+By+C=0.

The highest point of the tunnel entrance is 7 cm directly above the origin at ground level. The track, mounted on track bed PSPS, is centred about the origin. A train carrying containers which measured 2 cm wide may stack goods as high as 6.8 cm.

Solve for the constants A,B,CA,B,C, to 4 decimal places. Hence find the length of the widest part of the tunnel, to the nearest 0.1 cm. marks

Tunnel ellipse, track bed PS 4.3 cm, height 7 cm, container 2 cm wide and 6.8 cm high7 cmPOS4.3 cm6.8 cm2 cmOₓ
Question diagram 2
Tunnel ellipse, track bed PS 4.3 cm, height 7 cm, container 2 cm wide and 6.8 cm high7 cmPOS4.3 cm6.8 cm2 cmOₓ

Native tunnel and train diagram: 7 cm high tunnel, 4.3 cm track bed PS, 2 cm wide and 6.8 cm high container

[2008/ACJC/Prelim/II/2]

Solution

ABC

Fitting a conic

The elliptical entrance of a tunnel is modelled by x2+Ay2+By+C=0x^2+Ay^2+By+C=0. The highest point is 7 cm directly above the origin. The track bed has edges at (±2.15, 0)(\pm2.15,\,0). A train carrying a container 2 cm wide, stacked to 6.8 cm, must just fit, so the point (1, 6.8)(1,\,6.8) is on the ellipse. Solve for AA, BB, CC to 4 d.p.

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Hint

  1. Each of three boundary points gives one equation.

  2. (2.15,0)(2.15,0) has y=0y=0: it gives CC immediately.

  3. The other two points give equations in AA and BB.

Answer · ABC

A=0.6382,B=−3.8069,C=−4.6225A=0.6382,B=-3.8069,C=-4.6225 (4 d.p.).

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Three boundary points, three planes in (A, B, C)

Each point on the ellipse is a linear equation in the coefficients.

Open the interactive view to explore this geometry in 3D.

Width

Completing the square

Hence find the length of the widest part of the tunnel, to the nearest 0.1 cm.

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Hint

  1. Complete the square in yy to get the standard form x2+A(y−y0)2=Rx^2+A(y-y_0)^2=R.

  2. The widest part is on the line of symmetry y=y0y=y_0, where (y−y0)2=0(y-y_0)^2=0.

Answer · Width

6.4 cm (nearest 0.1 cm).

Question dimensions are cm; the margin note’s m units are typos.

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The fitted ellipse and its widest chord

The tunnel is the upper part of the ellipse. Its widest chord is on the line of symmetry y ≈ 2.98.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • Arch of the tunnel
  • Mirror lower half of the ellipse
  • Widest chord
Ideas to try
Is the ellipse widest where the container just fits? (No: the container is near the top.)