Understanding the Problem

We need to find the total length covered by the boundary of a regular hexagonal garden and its surrounding circular jogging track. This involves calculating the perimeter of the hexagon and the circumference of the circle separately, and then adding them together.

Calculating Hexagonal Garden Perimeter

A regular hexagon has 6 equal sides. The length of each side is given as 14 metres.

The formula for the perimeter of a regular hexagon is:

$ P_{hexagon} = 6 \times s $

Where '$s$' is the length of one side.

Substituting the given value:

$ P_{hexagon} = 6 \times 14 \text{ metres} $

$ P_{hexagon} = 84 \text{ metres} $

Calculating Circular Track Circumference

The jogging track is circular and surrounds the hexagon. The radius ('$r$') of the circle, measured from the centre of the hexagon, is given as 21 metres.

The formula for the circumference of a circle is:

$ C_{circle} = 2 \pi r $

We are given $\pi = \frac{22}{7}$.

Substituting the given values:

$ C_{circle} = 2 \times \frac{22}{7} \times 21 \text{ metres} $

$ C_{circle} = 2 \times 22 \times \frac{21}{7} \text{ metres} $

$ C_{circle} = 2 \times 22 \times 3 \text{ metres} $

$ C_{circle} = 44 \times 3 \text{ metres} $

$ C_{circle} = 132 \text{ metres} $

Finding the Sum of Perimeters

The question asks for the sum of the perimeters of both the hexagonal garden and the circular track.

$ \text{Sum of Perimeters} = P_{hexagon} + C_{circle} $

$ \text{Sum of Perimeters} = 84 \text{ metres} + 132 \text{ metres} $

$ \text{Sum of Perimeters} = 216 \text{ metres} $

Conclusion

The sum of the perimeters of the hexagonal garden and the circular jogging track is 216 metres.