Elizabeth needs to manufacture a cylinder with a height of three feet and a volume of 90 cubic feet. What will be the radius of the cylinder? Give your solution to two decimal places.
So we’re told that this cylinder needs to have a height of three feet. We’re also told what the volume of the cylinder needs to be, it’s 90 cubic feet. We need to work out what the radius of the cylinder needs to be in order to give the required volume. Let’s recall the formula for finding the volume of a cylinder. The volume is given by 𝜋𝑟 squared ℎ, where 𝑟 is the radius of the cylinder and ℎ is the height. In this question, we know two of these values: 𝑉 and ℎ. This means that we can substitute them into this formula in order to give an equation that we can solve for 𝑟. The volume, remember, is 90 cubic feet and the height is three. So we have the equation: 90 is equal to 𝜋 multiplied by 𝑟 squared multiplied by three.
Now we want to solve this equation for 𝑟. First of all, we need to divide both sides of the equation by three and by 𝜋. This gives 90 over three 𝜋 is equal to 𝑟 squared. Now both 90 and three can be divided by three, and so this fraction can be simplified. And if I just write the two sides of this equation the other way around, I have that 𝑟 squared is equal to 30 over 𝜋. Now I don’t want to know 𝑟 squared, I want to know 𝑟. So I now need to square root both sides of the equation. So I have that 𝑟 is equal to the square root of the fraction 30 over 𝜋. Now normally, when square rooting within an equation, we would use plus or minus the square root. But here, 𝑟 represents the radius of the cylinder, it’s a length; so it must be a positive value. Therefore, we’re only taking the positive square root.
Evaluating the square root using my calculator, tells me that 𝑟 is equal to 3.090193, and the decimal continues. Remember, the question asked us for a solution to two decimal places, so we need to round our answer.
Therefore, to two decimal places, we have that the radius of the cylinder is 3.09, and the units for this radius are feet.