# 10.5.2 Protecting an Old Tree

## 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?

### Answer

(a) The radius will be .

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

So, length of fencing = (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?

### Comment

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

### 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: . If the circumference increases by 1.5 cm a year, age of tree = years old. If the circumference increases by 2 cm a year, age of tree = 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?

### Answer

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

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

This means the diameter =

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

### Comment

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

### Answer

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

Root protection zone based upon trunk diameter = . Converting back into m, .

Since , 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