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The Central Problem of this unit

The central problem of this unit concerns two people who have planted an orchard on a circular lot.

Madie and Clyde want to know how long it will take before the trees grow so large that someone outside the orchard cannot see into the center of the orchard.

Answering this question will require you to study circles and coordinate geometry.

Be prepare to develop the formulas for the circumference and the area of a circle, as well as the distance and midpoint formulas, and learn to find the distance from a point to a line. Another theme of the unit is geometric proof.

Throughout this unit, you will be applying knowledge that you acquired in earlier units about similar triangles, trigonometry, and the Pythagorean theorem.

Identify yourself

An A⁺ student will review those concepts (similar triangles, trigonometry, and the Pythagorean theorem) before starting the unit.

An B  student will review those concepts (similar triangles, trigonometry, and the Pythagorean theorem) ONLY when they are needed.

An C student will  try to get by without studying or putting any effort on learning these concepts.

An F student would not care about these concepts.

Remember, the main unit question is...

Madie and Clyde want to know how long it will take before the trees grow so large that someone outside the orchard cannot see into the center of the orchard.

You can think of this question in term of these two more specific questions.

1. How big do the tree trunks have to become for the center  of the orchard to be a true hideout?

• How long will it take for the tree trunk to reach that size?

Take note of these two questions and share them with your class.

Look at the following diagram

Fig. 1:  An orchard with a radius of 3 trees.

Imagine flying an helicopter and looking down. You are watching an orchard.

What does it look like?

Look at the diagram above. Do you think that  this diagram can be representing a smaller orchard?

If each dot represent a tree, how many trees does this orchard have?

How many trees an orchard of radius 2 (two trees in each direction) will have?

How many in one with a radius of 5?

(Hint: make a similar diagram)

Notice that this diagram is representing an orchard with only 3 trees in each direction (North, South, East, and West.)

Here are some questions that you should be wondering about it.

Are all the trees of the same size?

Is the distance between each tree constant?

Are the trees growing a the same rate?

What size were the trees when they were planted?

Do you have a question of your own? If you do I will be glad to listed here with your name.

Mean while I will suggest that you read the reference pages (62-67) and share your learning with your class.

Note: Make sure that you use the terms circle, center, radius, and line of sight during your discussion in class.

Hint: Look for the mathematical definitions of these terms on the glossary.

Note: If you do what you did today I guarantee you a A+ in this course.

Keep it up!

Mr. Lora

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