10.5.2 Protecting an Old Tree

Activity Symbol Activity: Protecting an Old Tree

A real life example of how these formulas can be used is concerned with preserving old trees. In a local park, there is a large old beech tree. To help preserve old trees, the Woodland Trust (2004) recommends establishing a circular root protection zone with the tree at the center. The radius of this circle should be either 15 times the diameter of the tree trunk or 5 m beyond the canopy, whichever is greater.

To find the diameter of the tree trunk, it is easiest to measure the circumference of the tree trunk at chest height and then determine the diameter of the tree from that. The circumference can also be used to get an estimate of the age of the tree.

The circumference of a tree at chest height is 3.05 meters. The canopy has a radius of 12 m.

(a) What is the radius of the circle for the root protection zone, based on the radius of the canopy? If a wire fence is to be put up around this circular boundary, what length of fencing is needed?

Solution Symbol

Answer

(a) The radius will be 12 m plus five m equals 17 m.

The length of fencing required is the same as the circumference of the circle with a radius of 17 m.

circumference equation left hand side equals right hand side two multiplication pi postfix multiplication radius

So, length of fencing = two multiplication pi multiplication 17 m almost equals 106.8 m (to 1 decimal place).

So, a 107 m length of fencing should be sufficient.

(b) Beech trees are estimated to increase their trunk circumference by between 1.5 cm and 2 cm each year. Assuming this is true, how old is the beech tree if the increase was 1.5 cm a year and 2 cm per year?

Hint symbol

Comment

Convert the circumference of the beech tree from meters to centimeters (1 m = 100 cm).

Solution Symbol

Answer

[ If the circumference increases by 2 cm every year, can you work out the increase in the diameter? ] (b) Since the growth is given in cm, change the circumference measurement into cm first: 3.05 m equals three .05 multiplication 100 cm equals 305 cm. If the circumference increases by 1.5 cm a year, age of tree = 305 division 1.5 almost equals 203 years old. If the circumference increases by 2 cm a year, age of tree = 305 division two almost equals 153 years old. The growth will depend on the climate and other factors, so a reasonable estimate for the age of the tree is 150 to 200 years old.

(c) By drawing a “doing-undoing” diagram, work out a formula for the diameter of a circle if you know the circumference. What is the diameter of the tree trunk?

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Answer

(c) To find the circumference, you multiply the diameter by pi.

So, to find the diameter, you need to undo this step.

This means the diameter = 305 cm prefix division of pi almost equals 97 cm

(d) Based on your answers to parts (a) and (c), what radius should be used for the circular root protection zone?

Hint symbol

Comment

Calculate the root protection zone based upon the diameter of the tree trunk.

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Answer

(d) From part (a) the radius of the root protection zone = 17 m.

Root protection zone based upon trunk diameter = 15 multiplication 97 cm equals one comma 455 cm almost equals one comma 500 cm. Converting back into m, one comma 500 cm equals 1,500 division 100 m equals 15 m.

Since 17 greater than 15, the root protection zone should have a radius of 17 m.

Now we are going to move onto another property of shapes—their area.

10.5.1 Calculator Exploration: Introducing Pi