Video Transcript
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 ๐ด and ๐ต meet to form a circular cone of the greatest possible area. Determine the coneโs height to the nearest hundredth.
So, here, we have a circular sector which is folded up by making ๐ด and ๐ต meet to create a cone. Weโre told that the cone is of the greatest possible area. This means that ๐ด and ๐ต donโt overlap. Instead, we would simply have one line where ๐ด and ๐ต meet. Weโre asked here to find the height of the cone. We can call this any letter, but letโs define it as ๐ฅ. We need to use the information about this circular sector to help us find out this height.
Weโre told that the radius of the circular sector is 72 centimeters. And itโs 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, thatโs the value ๐ฅ, 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 itโs 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, letโs go back to our original sector. When we folded ๐ด and ๐ต together, this became the circumference of our new circle. We can find the circumference of a circle by calculating two times ๐ times the radius. But of course, our section ๐ด๐ต wasnโt 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 thereโs 360 degrees in total in a circle.
So, we can multiply this by two times ๐ times the radius. And since the radius is 72 centimeters, then we multiply our fraction by two times ๐ times 72. As for us to give our final answer to the nearest hundredth, we may assume weโre allowed to use a calculator, but itโs always worth simplifying a calculation if we can. Dividing our numerator and denominator by five gives us 55 over 72 times two times ๐ times 72. And we notice that we can eliminate 72 from the numerator and denominator of this calculation, leaving us with 55 times two ๐, which is 110๐ with units of centimeters.
So, now, we found this length in orange, which is ๐ด๐ต, and itโs 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 ๐๐, we can say that two ๐๐ is equal to 110๐. Canceling ๐ from both sides of the equation leaves us with two ๐ equals 110. And, therefore, ๐ 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 ๐. Plugging in the values then will give us 72 squared equals ๐ฅ squared plus 55 squared. Evaluating the squares, weโll have 5184 equals ๐ฅ squared plus 3025. Subtracting 3025 from both sides of the equation leaves us with 2159 equals ๐ฅ squared. Taking the square root of both sides and using our calculator to find decimal gives us ๐ฅ equals 46.46504 and so on.
Rounding this to the nearest hundredth means that we check our third decimal digit to see if itโs 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.
