A sector of a circle subtending an angle 300 degrees at the centre is used to form a cone with base radius 6cm. Find the (a.)radius of the circle (b.)volume of the cone (c.)area of the minor sector of the circle

It often helps to see a physical model of the problem

Cut out a circular sector with scissors and form it into a cone.
You will see that the arc length of the sector becomes the circumference of the circular base of the cone

if the cone is to have a radius of 6 cm, then its circumference is 12π cm

You should be familiar with the relation,
arc = rØ for a sector, where Ø is the sector angle in radians
300° to radians --->Ø/2π = 300/360---> Ø = 10π/6

12π = r(10π/6)
r = 12π(6/10π) = 7.2
So our sector must have a radius of 7.2 cm

b) If you look at your model you will see that 72 cm becomes the "slant" length of the cone, so using Pythagoras,
h^2 + 6^2 = 7.2^2 , where h is the height of the cone
h = √15.84

volume = (1/3) base x height
= (1/3)(36π)(√15.84)
= appr 150.04 cm^3

c) area of minor sector....
our needed sector is 5/6 of the circle, so the minor sector would be 1/6 of the area of the circle

area of minor sector = (1/6)π(7.2^2)
= appr 27.14 cm^2

check my arithmetic.

a.) Well, to find the radius of the circle, we can use some math wizardry. If the sector of the circle subtends an angle of 300 degrees at the center, that means the missing angle would be 60 degrees (since a full circle has 360 degrees). Now, let's figure out the circumference of the circle.

The circumference of a circle is given by the formula C = 2πr, where r is the radius. So, if the missing angle is 60 degrees, it means that 60/360 = 1/6th of the circle is missing. Therefore, the remaining sector is 5/6th of the original circle.

Now, let's set up a ratio to find the radius. We have the fraction 5/6 representing the remaining sector, and the circumference of the remaining sector should be 2π multiplied by the radius of the circle.

So, let's set up the equation:

(5/6) * C = 2πr

Since C = 2πr, we can substitute it in:

(5/6) * (2πr) = 2πr

The twos and pis on both sides cancel out:

(5/6) * r = r

Now, let's solve for r:

5/6 = 1

Unfortunately, 5/6 is not equal to 1, which means my math wizardry has failed us this time. It seems there might be a mistake in the problem statement, or maybe I need some extra clown makeup to figure this one out. Sorry about that!

b.) Since we couldn't find the radius of the circle, it's impossible to calculate the volume of the cone using the given information. However, if you come across a cone without any missing circles, I'll be more than happy to assist you with the calculations!

c.) Similar to part (b), without the radius of the circle, it's impossible to calculate the area of the minor sector using the given information. But hey, let's not cry over unseen circles – there are plenty of other fun geometry problems out there waiting for us!

To find the radius of the circle, we need to remember that the central angle of a sector is directly proportional to the radius of the circle.

(a.) Radius of the Circle:
The central angle of the sector is 300 degrees, which is 5/6 of the full circle (360 degrees). Let's denote the radius of the circle as 'r'.

Since the sector subtends an angle 300 degrees, we can set up the proportion:

300 degrees / 360 degrees = r / x

Simplifying the proportion:

5/6 = r / x

Cross-multiplication gives:

6r = 5x

Now we know that the base radius of the cone (formed from this sector) is 6 cm, so 'x' equals 6 cm.

Plugging in the value of 'x', we can solve for 'r':

6r = 5*6
6r = 30
r = 30/6
r = 5 cm

Hence, the radius of the circle is 5 cm.

(b.) Volume of the Cone:
The volume of a cone can be calculated using the formula:
Volume = (1/3) * π * r^2 * h

Since we know the base radius of the cone (r = 6 cm), we need to find the height of the cone.

The height of the cone can be found using the Pythagorean theorem, where the perpendicular side is the radius of the circle (5 cm) and the base is half the central angle (150 degrees).

Applying the Pythagorean theorem:

r^2 = h^2 + (base/2)^2

5^2 = h^2 + (150/2)^2

25 = h^2 + 75^2

25 = h^2 + 5625

h^2 = 5625 - 25

h^2 = 5600

h = √5600
h ≈ 74.83 cm

Now we can calculate the volume of the cone:

Volume = (1/3) * π * 6^2 * 74.83

Volume ≈ 933.03 cm^3

Hence, the volume of the cone is approximately 933.03 cm^3.

(c.) Area of the Minor Sector of the Circle:
The area of a sector can be calculated using the formula:
Area of sector = (θ/360) * π * r^2

Since the central angle of the minor sector is 300 degrees, we can calculate the area using:

Area of minor sector = (300/360) * π * 5^2

Area of minor sector = (5/6) * π * 25

Area of minor sector ≈ 54.96 cm^2

Hence, the area of the minor sector of the circle is approximately 54.96 cm^2.

To solve this problem, we'll use the formulas for the radius of a circle, volume of a cone, and area of a sector.

(a.) Finding the radius of the circle:
The angle subtended by the sector at the center of the circle is 300 degrees, which is 300/360 or 5/6 of the total angle of 360 degrees. The fraction of the circle's circumference taken up by the arc of the sector is equal to this fraction of the total circumference.

Let's denote the radius of the circle as "r". The circumference of the circle is given by the formula: C = 2πr, where π is a mathematical constant (approximately 3.14159).

The fraction of the circumference taken up by the arc of the sector is 5/6. Therefore, we can set up the equation:

(5/6) * (2πr) = length of the arc of the sector

The length of the arc of the sector is the perimeter of the base of the cone, which is given as 2π * 6 cm.

Simplifying the equation:

(5/6) * (2πr) = 2π * 6

(5/6) * r = 6

To find the value of "r", we can multiply both sides of the equation by (6/5):

r = (6/5) * 6

r = 7.2 cm

Therefore, the radius of the circle is 7.2 cm.

(b.) Finding the volume of the cone:
The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where "r" is the radius of the base of the cone and "h" is the height of the cone.

We know that the base radius is 6 cm. To find the height of the cone, we need to use trigonometry. The height is the perpendicular distance from the center of the base of the cone to the base.

Since the angle subtended by the sector is 300 degrees, the angle at the apex of the cone is half of this, which is 150 degrees. Using trigonometry, we can determine that the height of the cone is r * sin(150).

h = r * sin(150)

h = 7.2 * sin(150)

However, the sine of 150 degrees is -0.866, but the height cannot be negative. This means that the cone points downwards instead of upwards. In this case, we consider the absolute value of the height:

h = |7.2 * sin(150)|

h = 7.2 * 0.866

h ≈ 6.235 cm (rounded to 3 decimal places)

Now, we can calculate the volume of the cone:

V = (1/3) * π * (6^2) * 6.235

V ≈ 444.808 cm^3 (rounded to 3 decimal places)

Therefore, the volume of the cone is approximately 444.808 cm^3.

(c.) Finding the area of the minor sector of the circle:
The formula for the area of a sector is A = (θ/360) * π * r^2, where "θ" is the angle subtended by the sector at the center of the circle.

The angle of the sector is 300 degrees, so:

A = (300/360) * π * (7.2^2)

A = 5/6 * 3.14159 * 7.2^2

A = 120.739 cm^2 (rounded to 3 decimal places)

Therefore, the area of the minor sector of the circle is approximately 120.739 cm^2.

I hope this helps! Let me know if you have any further questions.