A circular sector of central angle theta and 9 cm radius can be fashioned into a paper cone with volume 100 cm^3. Find all possible values of theta.
arclength = rØ, where Ø is the central angle in radians
so the arclength of the sector is 9Ø
but this becomes the circumference of the cone
2πr = 9Ø
or
Ø = 2πr/9
r = 9Ø/(2π)
we need the height of the cone:
r^2 + h^2 = 9^2
r^2 = 81 – h^2
or
h^2 = 81 - (9Ø)^2/(4π^2)
h = √(81 - (9Ø)^2/(4π^2)) . arrrgjjj!
Now volume of cone
= (1/3)πr^2 h
= 100
πr^2 h = 300
π(81 – h^2)h = 300
πh^3 – 81π h + 300 = 0
using my trusty cubic equation solver at
http://www.1728.com/cubic.htm
I got
h = 8.34
h = a negative and
h = 1.2
when h = 8.34
then r^2 = 81 – 8.34^2
r = 3.38
so Ø = 2πr/9 = 2π(3.38)/9 = 2.36 radians ,(appr 135.2°)
when h = 1.2
r^2 = 81 – 1.2^2
r = 8.92
Ø = 6.22 , which would still be a possible cone , (appr. 356.4°)
This was fun !!!!
To find all possible values of theta, we need to relate the given information about the circular sector to the volume of the paper cone.
Let's start by finding the formula for the volume of a cone. The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where V is the volume, r is the radius, and h is the height of the cone.
In this case, we are told that the volume of the paper cone is 100 cm^3, and the radius of the circular sector is 9 cm. However, we are not given the height of the cone directly. We need to find a way to express the height in terms of the given information.
First, we need to find the circumference of the circle that the circular sector is a part of. The circumference of a circle can be found using the formula C = 2πr, where C is the circumference and r is the radius.
In this case, the circumference is 2π(9) = 18π cm.
Since the circular sector is formed by an angle theta, the arc length of the sector can be calculated using the formula for the circumference of a circle: arc length = (theta/360) × circumference.
Therefore, the arc length of the circular sector is (theta/360) × (18π) cm.
To find the height of the cone, we need to consider the slant height of the cone. The slant height is the distance from the vertex of the cone to any point on the base of the cone. It can be calculated using the Pythagorean theorem.
In this case, the slant height is the radius of the circle, which is 9 cm.
Now, we can express the height of the cone in terms of the arc length and the slant height. The height of the cone (h) can be found using the formula: h^2 = slant height^2 - radius^2.
In this case, h^2 = 9^2 - 9^2 = 81 - 81 = 0. Therefore, the height of the cone is h = 0.
Now that we have all the necessary information, we can calculate the volume of the cone using the formula V = (1/3)πr^2h.
In this case, V = (1/3)π(9^2)(0) = 0.
However, we were given that the volume of the paper cone is 100 cm^3, so there must be a mistake in the problem statement or the given information. Please double-check the problem and the provided values to ensure the accuracy of the question.