11. Infinitely many different sectors can be cut from a circular piece of paper with a 12-cm radius, and any such sector can be fashioned into a paper cone with a 12-cm slant height.
(a) Show that the volume of the cone produced by the 180-degree sector is larger than the volume of the cone produced by the 120-degree sector.
(b) Find a sector of the same circle that will produce a cone whose volume is even larger.
(c) Express the volume of a cone formed from this circle as a function of the central angle of the sector used to form it, then find the sector that produces the cone of greatest volume.
This is a question that needs to be visualized.
(I used to go to our teacher's staff room and bring back a cone paper cup from the water cooler and cut it open)
One has to realize that the arclength of the sector becomes the circumference of the circle of the cone, and the original radius of the sector becomes the slant height of the cone.
so for a 180° sector arclength = (1/2)(2π)12 = 12π cm
so the circumference of the base circle of the cone = 12π
2πr = 12π and the radius of the cone base is 6
then the height h is ...
h^2 + 6^2 = 12^2
h = √108 = appr. 10.3923
Volume of cone = (1/3)πr^2h = (1/3)π(36)√108 = 391.78
b)
since 120° is 1/3 of the rotation, the arclength would be (1/3) of 2π(12) or 8π
find the radius of the base circle as above, then the height, and finally the volume.
thanks
(a) Well, it's no surprise that the 180-degree sector cone will have a larger volume than the 120-degree one. After all, if you're giving the cone more room to stretch out, it's bound to get bigger! Just like when you eat an extra-large pizza compared to a small one, the volume always increases. So, yeah, the 180-degree sector cone wins in the volume department.
(b) If you want an even larger volume, you'll need a bigger slice of that circular paper. Imagine you're at a pizza party, and you want the biggest slice. You wouldn't settle for just a small piece, right? So, go ahead and take a 240-degree sector of that circle, and you'll get yourself a cone with an even larger volume. Bigger is better, my friend!
(c) Now, let's get a little mathematical here. The volume of a cone can be expressed as V = (1/3)πr^2h, where r is the radius and h is the height (which is equal to the slant height in this case). We'll call the central angle of the sector α.
To find the volume as a function of α, we need to figure out the radius and the height. Since the radius is fixed at 12 cm, we only need to find the height. And guess what? The height is just the slant height of the cone! So, the height can be given as h = 12 cm.
Now, let's plug these values into the volume formula. V = (1/3)π(12 cm)^2(12 cm). Simplifying that, we get V = 576π cm^3.
To find the sector that produces the cone with the greatest volume, we need to maximize the value of α. Since α represents the central angle, the largest possible value for α is 360 degrees (or 2π radians if you prefer). So, a 360-degree sector will give you the cone with the greatest volume. It's like having the whole pizza pie to yourself – can't get any better than that!
So, there you have it – the volume of the cone formed from the circle is given by V = 576π cm^3, and the sector that produces the cone of greatest volume is a 360-degree sector. Enjoy your cones, my friend!
To solve this problem, we will need to understand the properties of cones and how to calculate their volumes.
(a) Let's start by calculating the volume of the cone produced by the 180-degree sector.
The slant height of the cone is given as 12 cm. We can use the formula for the volume of a cone, which is V = (1/3) * π * r^2 * h, where r is the radius and h is the height.
Since the slant height is equal to the height in this case, the height of the 180-degree cone is also 12 cm.
Plugging these values into the formula, we get V(180-degree cone) = (1/3) * π * (12 cm)^2 * 12 cm.
Now let's calculate the volume of the cone produced by the 120-degree sector.
The height of this cone can be found using the Pythagorean theorem. Since the slant height is 12 cm, the height can be calculated as h = √(slant height^2 - radius^2) = √(12 cm)^2 - (12 cm/2)^2 = √144 cm^2 - 36 cm^2 = √108 cm^2 ≈ 10.39 cm.
Using the volume formula again, we have V(120-degree cone) = (1/3) * π * (12 cm)^2 * 10.39 cm.
Comparing the two volumes, we can see that V(180-degree cone) > V(120-degree cone).
(b) To find a sector that produces a cone with a larger volume than the previous ones, we need to increase the central angle.
Let's try a 240-degree sector. The height of this cone can be found using the same method as before. We find h = √(12 cm)^2 - (12 cm/2)^2 = √144 cm^2 - 36 cm^2 = √108 cm^2 ≈ 10.39 cm.
Using the volume formula, we have V(240-degree cone) = (1/3) * π * (12 cm)^2 * 10.39 cm.
Comparing these volumes, we can see that V(240 degree cone) > V(180-degree cone).
(c) Now let's express the volume of a cone formed from this circle as a function of the central angle of the sector used to form it.
The height of the cone can be expressed as h = slant height * sin(θ/2), where θ is the central angle.
Using the volume formula once again, we have V(θ-degree cone) = (1/3) * π * (12 cm)^2 * (12 cm * sin(θ/2)).
To find the sector that produces the cone of greatest volume, we will need to maximize this volume function.
One way to approach this is by finding the maximum value of sin(θ/2), which occurs when θ/2 = 90 degrees or θ = 180 degrees.
Therefore, the sector that produces the cone of greatest volume is the 180-degree sector.
To summarize:
(a) The volume of the cone produced by the 180-degree sector is larger than the volume of the cone produced by the 120-degree sector.
(b) The sector that produces an even larger volume is the 240-degree sector.
(c) The volume of a cone formed from this circle as a function of the central angle is V(θ-degree cone) = (1/3) * π * (12 cm)^2 * (12 cm * sin(θ/2)). The sector that produces the cone of greatest volume is the 180-degree sector.