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# Activity: The Big Bang Theory – The sweet spot

## Purpose

To investigate conic sections in real life situations.

## Achievement objectives

M8-1 Apply the geometry of conic sections

## Indicators

• Demonstrates understanding of conic sections, including circles, ellipses, parabolas, hyperbolas, including links to transformations and definitions of loci.
• Connects equations in parametric and Cartesian form and the graphs of conic sections, including understanding the standard equation of each type of conic section.
• Identifies and uses key features of conic sections, such as vertices, foci, asymptotes, and intercepts.

## Specific learning outcomes

Students will be able to:

• draw and discuss conic sections
• apply an understanding of conic sections to real life situations.

## Planned learning experiences

### Introduction: Whispering galleries

A whispering gallery is usually constructed in the form of an ellipsoid, with an accessible point at each focus. When a visitor stands at one focus and whispers, the line of sound emanating from this focus reflects directly to the dish/focus at the other end of the room, and to the other person. Circular whispering galleries may provide "communication" from any part on the circumference to the diametrically-opposite point on the circumference.

Visit this website for an introduction to whispering galleries:

Invite students to research examples of whispering galleries around the world.

Invite students to model vertical cross sections of whispering galleries with conic sections (ellipses) given the two foci of the whispering gallery. Introduce physical constraints such as height and length of the room to limit the possible models.

### The sweet spot

This link is to a YouTube clip from the television series the Big Bang Theory (Season 2, Episode 9). It may be more appropriate to rent or buy the DVD to show in class. The YouTube link may not be maintained.

The sweet spot is a term used by recording engineers to describe the intersection of focal points from conic shaped speakers, in this position an individual is fully capable of hearing the stereo audio mix the way it was intended to be heard by the mixer.

Assuming there are two speakers at the front of the cinema which are pieces of conic sections, experiment with different speaker shapes and sizes to place the sweet spot as close to Sheldon’s seat as possible. Sheldon’s seat is 8m from the screen and level with the height of the centre of the speakers. The conic section should model either the vertical or horizontal cross-section of the speakers.

• Which is the best conic section to place the sweet spot at Sheldon? Write the equations of these graphs to model the situation and draw an appropriate diagram.
• Consider the variables in this scenario. What factors would a sound engineer take into account when setting up a cinema?

### Possible adaptations to the activity

Links could be made to maximum and minimum points, differentiation and volumes of revolution when considering properties of the solids generated by rotating the conic models found in the previous investigations.

• Architecture
• Art and design

## Extension/enrichment ideas

• Check out some of the links below to conic sections in real life. Design the blue prints for your conic building.
• Find dimensions, surface areas and volumes for the purposes of costing.
• Investigate materials used for constructing such buildings and find some interesting mathematical problems that architects and builders may have encountered in their design and/or construction.

Cybertecture_1

Cybertecture_2

## Planned assessment

This teaching and learning activity could lead towards assessment in the following achievement standard:

## Spotlight on

### Pedagogy

• Creating a supportive learning environment
• valuing student contributions
• respecting and valuing the mathematics and cultures that students bring to the class room.

### Key competencies

• Thinking:
• Students select appropriate methods and strategies when solving problems.
• Students design investigations, explore and use patterns and relationships in data and they predict and envision outcomes.
• Using language, symbols and texts:
• Students use symbols and diagrams to solve problems.
• Relating to others:
• Students work in groups, they debate solutions, negotiate meaning and communicate thinking.
• Managing self:
• Students are prepared to take risks, make decisions, and persevere.

### Values

Students will be encouraged to value:

• Innovation, inquiry and curiosity by thinking critically, creatively, and reflectively

### Māori/Pasifika

The Design House Mahina

• Home Designs unique beach slope, here The Design House Mahina – Futuristic Mansion with a film of James Bond in Kawau Island, north of Auckland. The Mahina – Māori for the moon – has been designed in a crescent-shaped curve with the glass ceiling to floor and a modern white decor.

## Connections

Last updated September 10, 2018