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Themed maths

Spring egg geometry for IB Maths SL and HL

A ready-to-teach lesson for the spring term: model an egg with an ellipse, then with a curve that is fatter at one end, and find areas and volumes. A 5-minute starter, a 35-minute main activity, an HL extension and full worked answers.

Level
IB Maths SL and HL (AA and AI)
Time
40 minutes, plus a 10-minute extension
Topics
Volumes of revolution (HL; SL with a GDC); Area under a curve; Optimisation with a GDC; Similar shapes and scale factors
Equipment
The starter is non-calculator. A GDC is needed for the main activity.

Download student sheet (PDF)Answers (PDF)

Suggested timings

PartTimeWhat
Starter5 minQuick questions on the board
Main: task A11 minAn egg as an ellipse
Main: task B12 minThe volume of an egg
Main: task C12 minA more egg-like egg
Extension10 minFast finishers or homework

Starter (5 minutes)

No GDC.

  1. Find the volume of a sphere of radius 3, in terms of π.
  2. Solve x2/25 = 1.
  3. Work out ∫02 x2 dx.
  4. Two similar solids have lengths in the ratio 1 : 2. What is the ratio of their volumes?

Main activity (35 minutes)

Task A: An egg as an ellipse (11 min)

The outline of an egg is modelled by x2/9 + y2/4 = 1, in centimetres.

  1. Write down the length and the width of the egg.
  2. Show that the top half of the outline is y = 2√(1 − x2/9). Find y when x = 1.5, exactly and to 3 significant figures.
  3. Use your GDC to find the area inside the ellipse (twice the area under the top half). Show that it equals 6π, to 3 significant figures.

Task B: The volume of an egg (12 min)

The top half of the ellipse, y = 2√(1 − x2/9), is rotated 2π about the x-axis to make a solid egg. (HL by hand; SL with a GDC.)

  1. Find the volume of the egg, exactly and to 3 significant figures.
  2. A sphere has the same length as the egg (radius 3 cm). What fraction of the sphere’s volume is the egg?
  3. A hollow chocolate egg has outside surface given by a = 3, b = 2 and inside surface a = 2.8, b = 1.8 (cm), where the volume of such an egg is 4⁄3πab2. Find the volume of chocolate, to 3 significant figures.

Task C: A more egg-like egg (12 min)

Real eggs are fatter at one end. A better model is y2 = (4/9)(9 − x2)(1 + x/6), for −3 ≤ x ≤ 3. Plot y = ±√((4/9)(9 − x2)(1 + x/6)) on your GDC.

  1. Where does the egg meet the x-axis?
  2. Find the widest point: the value of x where y is greatest, and that y, each to 3 significant figures.
  3. The egg is rotated 2π about the x-axis. Show that its volume is exactly the same as the ellipse egg in task B.

Extension (10 minutes)

Extension 1 is for everyone; extension 2 is HL.

  1. Every length of the egg in task C is multiplied by 1.5. Find its new volume, exactly and to 3 significant figures.
  2. (HL) The part of the ellipse x2/9 + y2/4 = 1 with x ≥ 0 is rotated 2π about the y-axis instead. Find the volume.

For teachers

Teacher notes and full worked answers
  • Bring in real eggs and a ruler: students can measure the length and width and compare their own model with the egg.
  • Task C (c) is a lovely surprise: the lopsided egg has exactly the same volume because the extra factor adds an odd function.
  • AI SL students can do every part with the GDC’s numerical integration; AA HL students should do task B by hand.

Starter

  1. 36π
    • 4⁄3π × 27 = 36π
  2. x = 5 or −5
    • x2 = 25
  3. 8/3
    • [x3/3]02 = 8/3
  4. 1 : 8
    • Volume scale factor = 23 = 8

Task A: An egg as an ellipse

  1. 6 cm long, 4 cm wide
    • y = 0 gives x = ±3; x = 0 gives y = ±2.
  2. √3 ≈ 1.73
    • y2 = 4(1 − x2/9), and y ≥ 0 on the top half.
    • x = 1.5: y = 2√(1 − 0.25) = 2√0.75 = √3 = 1.732…
  3. 18.8 cm2 = 6π
    • 2∫−33 2√(1 − x2/9) dx = 18.849…
    • 6π = 18.849…: in general the area of an ellipse is πab.

Task B: The volume of an egg

  1. 16π ≈ 50.3 cm3
    • V = π∫−33 4(1 − x2/9) dx = 4π[x − x3/27]−33 = 4π × 4 = 16π.
    • 16π = 50.26…
  2. 4/9
    • Sphere: 36π. 16π/36π = 4/9
  3. 12.3 cm3
    • Outside: 4⁄3π × 3 × 4 = 16π. Inside: 4⁄3π × 2.8 × 3.24 = 12.096π.
    • 16π − 12.096π = 3.904π = 12.26…

Task C: A more egg-like egg

  1. x = 3 and x = −3
    • y = 0 when 9 − x2 = 0 (or 1 + x/6 = 0, which gives x = −6, outside the domain).
  2. x = 0.646, y = 2.06
    • Maximise (9 − x2)(1 + x/6): the derivative is zero when x2 + 4x − 3 = 0, so x = −2 + √7 = 0.6457…
    • y = √((4/9)(9 − 0.417)(1.1076)) = 2.0555…
  3. 16π
    • V = π∫−33 (4/9)(9 − x2)(1 + x/6) dx = (4π/9)[∫(9 − x2) dx + (1/6)∫x(9 − x2) dx].
    • The second integral is of an odd function over −3 to 3, so it is 0. The first is 36.
    • V = (4π/9) × 36 = 16π

Extension

  1. 54π ≈ 170 cm3
    • Volume scale factor 1.53 = 3.375.
    • 16π × 3.375 = 54π = 169.6…
  2. 24π
    • x2 = 9(1 − y2/4).
    • V = π∫−22 9(1 − y2/4) dy = 9π[y − y3/12]−22 = 9π × 8/3 = 24π

The same answers are in the answers PDF. Every answer was recomputed by computer before publishing.

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