A cylinder has a volume of 1241 centimeters cubed and a height of 13 centimeters. The radius of this cylinder is equal to the radius of a sphere. Find the volume of the sphere to the nearest centimeter cubed.
So, in this question, we’re considering two three-dimensional objects, a cylinder and a sphere. We’re not given a diagram, but it can be very helpful to draw a sketch for ourselves and fill in the information that we’re given. So, let’s consider our cylinder. We know that it’s got a height of 13 centimeters and we know that the volume is 1241 centimeters cubed. We’re not told what the radius is, but we do know that it’s equal to the radius of the sphere.
So, let’s call the radius of both of these 𝑟. Let’s see if we have enough information to work out the radius 𝑟, given we know the volume and the height of our cylinder. We can use the formula for the volume of a cylinder, which is equal to 𝜋𝑟 squared ℎ, where 𝑟 is the radius and ℎ is the height. So, let’s start by writing our formula and then plugging in any values that we know. Since the volume is 1241 and the height is 13, this will give us 1241 equals 𝜋𝑟 squared times 13.
And when we have 𝜋, and 𝑟 squared, and 13 all written next to each other, that just means that we’re going to multiply them. And it doesn’t matter which way round we multiply them together. And a reminder that this symbol 𝜋 simply represents a number 3.14 and so on. But we can use our calculator in a minute to get the value for the answer. So, now that we have 1241 equals 13𝜋𝑟 squared, what we really want to do is to work out 𝑟. So, we need to rearrange to get 𝑟 by itself.
So, since we have 𝑟 squared multiplied by 13𝜋, to rearrange it, we’re going to start by doing the inverse of multiplying by 13𝜋. And we’re going to divide both sides of our equation by 13𝜋. This will give us 1241 over 13𝜋 equals 𝑟 squared. We can do that in two steps instead by dividing by 13 and then by dividing by 𝜋. But we will end up with the same answer here. So, now that we have 𝑟 squared equals, we need to work out how to get 𝑟 equals.
And since 𝑟 was squared, then we need to do the inverse which is to take the square root of both sides. This will give us the square root of all of 1241 over 13𝜋 equals 𝑟, which we can write the other way round as 𝑟 equals the square root of 1241 over 13𝜋. So, at this point, we’re going to use this value for 𝑟 to work out the volume of the sphere. So, the best thing is to keep it in this square root form. But you can also use your calculator to work it out as a numerical value.
In this case, the radius would be equal to 5.5123816 and so on centimeters. And we’ll try and keep as many decimal places as we can. So, let’s see then if we can work out the volume of the sphere. We’re going to use the formula the volume of a sphere equals four-thirds 𝜋𝑟 cubed. So, all we need to know is the value for the radius. So, now, we write our formula and we plug in the value for the radius that we know.
This will give us the volume of a sphere equals four-thirds 𝜋 times the cube of the square root of 1241 over 13𝜋. Or if we were using the decimal equivalent for our radius, we would have the volume equals four-thirds 𝜋 times 5.5123816 cubed. So, now, we need to use a calculator to work out the value for the volume. And this can often be the trickiest part. If we are working from our blue line, it can be helpful to start to see if we can get this value 5.5123 and so on for the radius. Then, it’s important that we cube only that part, the radius, before multiplying it by our four-thirds and by the 𝜋.
So, when we have evaluated this using a calculator, this will give us 701.67243 and so on centimeters cubed. But since we were asked to find the volume to the nearest centimeters cubed, we’re going to need to round. In this case, this will be equivalent to rounding to the nearest whole number. So, we know that our answer will either be 701 or 702. So, we check if the column on the right has a digit that is five or more. And since it has the digit six, this means that our value will round up to 702. And so, our final answer is the volume of the sphere is 702 centimeters cubed, to the nearest centimeter cubed.