### Video Transcript

A landscape gardener decides that
he wants to design a lawn split into a series of sectors with circular patios laid
into the grass, as shown in the given figure. The circular lawn will be split
into six equal sectors, each with a radius of eight meters. The lines ππ΄ and ππ΅ are both
tangent to the circle and the arc π΄π΅ touches the circle at a single point.

This question is broken into three
parts. Letβs consider the first part.

Work out the area of sector
ππ΄π΅. Give your answer in terms of
π.

Recall that to find the area of a
sector, we use the formula one-half times the radius squared times π, where π is
the angle of the sector given in radians. Weβll need to convert 60 degrees to
radians. And we can do that by multiplying
60 by π over 180. 60 over 180 reduces to one over
three, which makes 60 degrees π over three radians. Now, we can plug in what we know
into our formula, and weβll have one-half times eight squared times π over
three. Eight squared is 64. 64 times one-half is 32. The area of the sector is then 32π
over three. Remember, our radius was given to
us in meters. And that makes the units for the
area square meters.

For part two, the gardener needs to
calculate the radius of the circular patio. Using trigonometric ratios,
calculate the radius of the patio. Give your answer as a fraction.

There are two triangles weβll need
to consider to find the radius of the circular patio. Remember that the angle between a
tangent to a circle and that circleβs radius is π over two radians, or 90
degrees. If we let the center of the circle
be point π, we can sketch a right-angle triangle whose hypotenuse runs from π to
π. If we extend our line ππ up
further to point πΆ, we can say that the line ππΆ must be equal to eight meters
since it is a line thatβs drawn from the center to a point on the circumference of
the lawn. The line segment ππΆ would also be
a radius of the inner circle. We can say then that the line
segment ππ plus the line segment ππΆ must equal eight. And the line segment ππΆ is a
radius of that circular patio. Solving for π here, we find that
the radius is equal to eight minus ππ.

To solve the value of the radius,
weβll need to find a second equation that expresses π in terms of ππ. We can use this smaller triangle to
do that. Here is ππ and hereβs the π we
want to write an equation for. We have an opposite side length and
hypotenuse. Therefore, weβll use the sine
ratio. Remember, when we had 60 degrees,
we were dealing with π over three. Weβve cut that angle in half, which
means weβre dealing with an angle of π over six. And we can say that the sin of π
over six is equal to π over ππ. sin of π over six equals
one-half. If we multiply both sides of the
equation by two and then multiply both sides of the equation by ππ, we find that
ππ equals two π.

In our first equation, we can
substitute two π in place of ππ. If we add two π to both sides, we
get that three π equals eight. Therefore, the radius is
eight-thirds. And remember, weβre measuring in
meters. So, we can say the radius of the
patio will be eight-thirds meters.

For the final part of this
question, calculate the total area of grass in one sector. Give your answer in terms of π in
its simplest form.

To find the area of grass, we need
to take the area of the whole sector and subtract the portion that is covered with
patio. We found the area of sector ππ΄π΅
in part one to be 32π over three. The area of the patio will be equal
to ππ squared as it is a circle. The radius of the circular patio we
found in part two to be eight-thirds. Eight-thirds squared is 64 over
nine, which makes the area of the patio 64π over nine. 32π over three minus 64π over
nine is equal to 32π over nine, which makes the total area covered in grass 32π
over nine square meters.