Determine the volume of the given right circular cone in terms of 𝜋.
To find the volume of a cone, we actually use the pyramid formula because technically a cone is a circular pyramid. This formula is one-third times the area of the base times the height. And since our base is a circle, instead of 𝐵, we can replace it with 𝜋𝑟 squared. So the volume of a cone is one-third times 𝜋 times 𝑟 squared, where 𝑟 is the radius of the circle, times the height of the cone.
Looking at our diagram, we can see that the height is forty-eight centimeters but we do not have the radius. However, we can solve for the radius using this right triangle. We can use the Pythagorean theorem.
Before we use the theorem, let’s see what we know about this right triangle. The height of this triangle is the exact same as the height of the pyramid. So it’s forty-eight centimeters. The longest side of this right triangle is gonna be sixty centimeters because this is a property of the cone. If the slant height is sixty on one side, then it’s also gonna be sixty on the other side.
Now we can use the Pythagorean theorem to solve for the radius. The Pythagorean theorem states: The sum of the squares of the two shorter sides will be equal to the square of the longest side. Lets go ahead and plug these in. And instead of the radius equals a question mark, let’s just use 𝑥.
𝑥 squared plus forty-eight squared equals sixty squared. 𝑥 squared is equal to 𝑥 squared. Forty-eight squared is equal to two thousand three hundred and four. And sixty squared is equal to three thousand six hundred. Now let’s subtract two thousand three hundred and four from both sides of the equation. So we get that 𝑥 squared is equal to one thousand two hundred and ninety-six. And our final step would be to square root both sides of the equation. And we get that 𝑥 equals thirty-six, which is our radius.
Now that we know the radius is thirty-six centimeters and the height is forty-eight centimeters, we can plug it in to our volume of a cone formula. So now we have volume is equal to one-third times 𝜋 times thirty-six centimeters, which we need to square, times forty-eight centimeters. Let’s go ahead and square thirty-six centimeters. Thirty-six centimeters squared is equal to one thousand two hundred and ninety-six centimeters squared.
Notice we could’ve made this question a little bit shorter because when we were solving for the radius, we could’ve just found the radius squared to equal one thousand two hundred and ninety-six and plug it into our formula. Nevertheless, we can keep going where we’re gonna be multiplying all of our numbers together. Now notice it said to leave our answer in terms of 𝜋. That means we don’t wanna multiply by 𝜋. We will wanna keep it in our answer as just 𝜋.
So let’s go ahead and multiply our two large numbers together, and then we will multiply by the one-third. which results in sixty-two thousand two hundred and eight centimeters cubed. This is because we multiplied centimeters squared times centimeters. When you multiply, you add your exponents, and a volume should be in centimeters cubed. So this is great. So now you’ll take sixty-two thousand two hundred and eight and multiply it by one-third.
Leaving your answer in terms of 𝜋, we get twenty thousand seven hundred and thirty-six 𝜋 centimeters cubed, or also it can be heard as cubic centimeters.