A piece of paper in the shape of a circular sector having a radius of 72 centimeters and an angle of 275 degrees is folded in a way so that the points 𝐴 and 𝐵 meet to form a circular cone of the greatest possible area. Determine the cone’s height to the nearest hundredth.
So, here, we have a circular sector which is folded up by making 𝐴 and 𝐵 meet to create a cone. We’re told that the cone is of the greatest possible area. This means that 𝐴 and 𝐵 don’t overlap. Instead, we would simply have one line where 𝐴 and 𝐵 meet. We’re asked here to find the height of the cone. We can call this any letter, but let’s define it as 𝑥. We need to use the information about this circular sector to help us find out this height.
We’re told that the radius of the circular sector is 72 centimeters. And it’s very important to note that this will be different from the radius of the circular section of the cone because we have in fact created a smaller circle on the base of the cone. When, in fact, we folded this circular sector together, then the radius of the circular sector becomes the slant height of the cone. We may also notice that we have created a triangle within this cone. And not just any triangle, it will be a right triangle.
In a right triangle, we could apply the Pythagorean theorem, which tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides. In order to find that the height of this cone, that’s the value 𝑥, we do need to work out this other missing length, which will, of course, be the radius of the this circle, and, of course, remembering that it’s not 72 centimeters. That was just the radius of the original circular sector.
We can work out the radius by finding the circumference of this circle. And how do we do that? Well, let’s go back to our original sector. When we folded 𝐴 and 𝐵 together, this became the circumference of our new circle. We can find the circumference of a circle by calculating two times 𝜋 times the radius. But of course, our section 𝐴𝐵 wasn’t the circumference of the whole circle, but rather a fraction of it. This fraction would be the angle of 275 degrees over 360 degrees as there’s 360 degrees in total in a circle.
So, we can multiply this by two times 𝜋 times the radius. And since the radius is 72 centimeters, then we multiply our fraction by two times 𝜋 times 72. As for us to give our final answer to the nearest hundredth, we may assume we’re allowed to use a calculator, but it’s always worth simplifying a calculation if we can. Dividing our numerator and denominator by five gives us 55 over 72 times two times 𝜋 times 72. And we notice that we can eliminate 72 from the numerator and denominator of this calculation, leaving us with 55 times two 𝜋, which is 110𝜋 with units of centimeters.
So, now, we found this length in orange, which is 𝐴𝐵, and it’s also the circumference of the circle on the base of the cone. We can use this now to find the radius of the circle on the base of the cone. As we know that the circumference is equal to two 𝜋𝑟, we can say that two 𝜋𝑟 is equal to 110𝜋. Canceling 𝜋 from both sides of the equation leaves us with two 𝑟 equals 110. And, therefore, 𝑟 is equal to half of 110, which is 55 centimeters.
So, now, if we look at the right triangle we created in the cone, we have two values that we know and one missing value, the height of the cone. We can finally apply the Pythagorean theorem, recalling that 72 centimeters is our hypotenuse 𝑐. Plugging in the values then will give us 72 squared equals 𝑥 squared plus 55 squared. Evaluating the squares, we’ll have 5184 equals 𝑥 squared plus 3025. Subtracting 3025 from both sides of the equation leaves us with 2159 equals 𝑥 squared. Taking the square root of both sides and using our calculator to find decimal gives us 𝑥 equals 46.46504 and so on.
Rounding this to the nearest hundredth means that we check our third decimal digit to see if it’s five or more. And as it is, then our answer rounds up to 46.47. And as this is a height, then the units will be the length units of centimeters. So, our final answer is that the height of the cone is 46.47 centimeters.