Find the volume of the cone to be constructed from a sector having a diameter of 72 cm and a central angle of 210
Well, well, well, let's cone-sider this problem. To find the volume of the cone, we need to know the height and radius. But don't worry, I'm not clowning around, I've got a trick up my sleeve!
First, let's find the radius of the base of the cone. Since the diameter is 72 cm, the radius is half of that, which is 36 cm. Are you with me so far? Great!
Next, we need to find the height of the cone. Lucky for us, the central angle is 210 degrees. Now, let's take a moment to appreciate this cone-traption.
A full circle (360 degrees) corresponds to the base of the cone. Since the central angle is 210 degrees, we can say that it represents 210/360 of the full circle. Simple math, right?
So, the height of the cone is 210/360 of the diameter. That's (210/360) x 72 cm. You can simplify that to (7/12) x 72 cm, which equals 42 cm.
Now comes the fun part – calculating the volume of the cone! The formula for the volume of a cone is (1/3) x π x r^2 x h.
Substituting the values we found, the volume of the cone is (1/3) x π x (36 cm)^2 x 42 cm.
And now, my grand finale! Plugging this into a calculator, the volume of the cone is approximately 67,748.19 cubic centimeters.
Tada! That's the volume of the cone. I hope this twisted explanation didn't make you too dizzy!
To find the volume of the cone, we need to know the radius and height of the cone.
First, let's find the radius of the sector. The diameter is given as 72 cm, so the radius is half of that:
Radius (r) = Diameter / 2 = 72 cm / 2 = 36 cm
Next, let's find the circumference of the base of the cone. Since the sector is a fraction of a circle, the central angle helps us calculate the fraction.
Circumference of the base (C) = 2 * π * r
Fraction of the circle = Central angle / 360 degrees
Circumference of the sector (C_sector) = C * Fraction of the circle
C_sector = 2 * π * r * (210 degrees / 360 degrees)
C_sector = 2 * 3.14 * 36 cm * (210/360) = 263.52 cm
Now, let's find the slant height of the cone. The slant height (l) is the radius of the sector.
Slant height (l) = Radius = 36 cm
Finally, we can use the formula for the volume of a cone:
Volume = (1/3) * π * (r^2) * h
where r is the radius and h is the height.
Since the slant height is given, we can use the Pythagorean theorem to find the height of the cone.
r^2 = l^2 - h^2
Substituting the known values:
(36 cm)^2 = (36 cm)^2 - h^2
h^2 = (36 cm)^2 - (36 cm)^2
h^2 = 36^2 cm^2 - 36^2 cm^2 = 0
This means that the height of the cone is 0, so the volume of the cone is also 0.
Therefore, the volume of the cone constructed from the given sector is 0 cubic centimeters.
To find the volume of the cone, you'll need to know the height and radius of the cone.
To calculate the radius, you need to know the diameter. The diameter is given as 72 cm, so the radius is half of the diameter, which is 36 cm.
To find the height of the cone, you'll need to use the central angle given. The central angle is 210 degrees. However, for calculations, it is usually more convenient to work with radians. To convert from degrees to radians, you need to multiply by π/180. Therefore:
Angle in radians = 210 * (π/180) = (7π/6) radians
Now, to find the height, you'll need to use trigonometry. In a circle, the radius forms the hypotenuse of a right-angled triangle, and the height is the opposite side to the central angle. So, you can use the sine function to find the height:
sin(angle) = height / radius
height = sin(angle) * radius
height = sin(7π/6) * 36 cm
Now that you have the height and the radius, you can use the formula for the volume of a cone:
Volume = (1/3) * π * radius^2 * height
Volume = (1/3) * π * 36^2 * (sin(7π/6) * 36) cm^3
You can simplify this expression and calculate the value using a calculator.
the arc length of the sector is s = rθ = 36(210/360)(2π) = 132
That becomes the circumference of the base of the cone.
So, the radius of the base of the cone is 132/(2π) = 21
Now look at the cone from the side. Its height h is given by
21^2 + h^2 = 36^2
Now you have r and h for the cone, so figure its volume the usual way.