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 𝑂𝐴 and 𝑂𝐵 are both
tangent to the circle and the arc 𝐴𝐵 touches the circle at a single point.
This question is broken into three
parts. Let’s consider the first part.
Work out the area of sector
𝑂𝐴𝐵. 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 𝜃, where 𝜃 is
the angle of the sector given in radians. We’ll 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 we’ll 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 we’ll need
to consider to find the radius of the circular patio. Remember that the angle between a
tangent to a circle and that circle’s radius is 𝜋 over two radians, or 90
degrees. If we let the center of the circle
be point 𝑃, we can sketch a right-angle triangle whose hypotenuse runs from 𝑂 to
𝑃. If we extend our line 𝑂𝑃 up
further to point 𝐶, we can say that the line 𝑂𝐶 must be equal to eight meters
since it is a line that’s drawn from the center to a point on the circumference of
the lawn. The line segment 𝑃𝐶 would also be
a radius of the inner circle. We can say then that the line
segment 𝑂𝑃 plus the line segment 𝑃𝐶 must equal eight. And the line segment 𝑃𝐶 is a
radius of that circular patio. Solving for 𝑟 here, we find that
the radius is equal to eight minus 𝑂𝑃.
To solve the value of the radius,
we’ll need to find a second equation that expresses 𝑟 in terms of 𝑂𝑃. We can use this smaller triangle to
do that. Here is 𝑂𝑃 and here’s the 𝑟 we
want to write an equation for. We have an opposite side length and
hypotenuse. Therefore, we’ll use the sine
ratio. Remember, when we had 60 degrees,
we were dealing with 𝜋 over three. We’ve cut that angle in half, which
means we’re dealing with an angle of 𝜋 over six. And we can say that the sin of 𝜋
over six is equal to 𝑟 over 𝑂𝑃. 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 𝑂𝑃, we find that
𝑂𝑃 equals two 𝑟.
In our first equation, we can
substitute two 𝑟 in place of 𝑂𝑃. If we add two 𝑟 to both sides, we
get that three 𝑟 equals eight. Therefore, the radius is
eight-thirds. And remember, we’re 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 𝑂𝐴𝐵
in part one to be 32𝜋 over three. The area of the patio will be equal
to 𝜋𝑟 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.
