Question Video: Solving Volume Problems Involving Spheres and Cylinders Mathematics • Second Year of Preparatory School

Video Transcript

The sphere and cylinder in the given figure are to be constructed with equal volumes. Work out a formula for 𝑟 in terms of ℎ. Given that the height of the cylinder needs to be 18 inches, find the volume of the two solids. Give your answer to two decimal places.

So here we have a sphere and a cylinder. And given that we’re discussing the volumes in this question, it can be helpful to note down the formulas for the volumes of both of these shapes. The volume of a sphere is equal to four-thirds 𝜋𝑟 cubed, where 𝑟 is the radius. The volume of a cylinder is equal to 𝜋𝑟 squared ℎ, where ℎ is the height. Notice of course that the 𝜋𝑟 squared refers to the area of the circle at the base of the cylinder. Given that we don’t know any values for 𝑟 or ℎ, we can put the volume of the sphere is equal to four-thirds 𝜋𝑟 cubed. And the volume of the cylinder is 𝜋𝑟 squared ℎ.

We’re given in the question that these two shapes have equal volumes, which means that we can set four-thirds 𝜋𝑟 cubed equal to 𝜋𝑟 squared ℎ. Notice that this will help us work out our first question because we now have an equation in terms of 𝑟 and ℎ. Looking at our equation, we can cancel the 𝜋 from both sides, leaving us with four-thirds 𝑟 cubed equal to 𝑟 squared ℎ. In order to write 𝑟 in terms of ℎ, we want to have all terms involving 𝑟 on the same side of the equation. Therefore, subtracting 𝑟 squared ℎ from both sides gives us four-thirds 𝑟 cubed minus 𝑟 squared ℎ equals zero.

We can then factor this by taking out 𝑟 squared as a common factor. So the first term in our parentheses will be four-thirds 𝑟 since 𝑟 squared times four-thirds 𝑟 gives us four-thirds 𝑟 cubed. And the second term in parentheses will be negative ℎ since 𝑟 squared times negative ℎ gives us negative 𝑟 squared ℎ.

To solve for 𝑟, we can recall that if we have two values 𝑎 and 𝑏 which multiply to give zero, then 𝑎 equals zero or 𝑏 equals zero. In this case, we have 𝑟 squared equals zero or four-thirds 𝑟 minus ℎ equals zero. So when 𝑟 squared is equal to zero, then 𝑟 must be equal to zero since the square root of zero is zero. And when four-thirds 𝑟 minus ℎ equals zero, then adding ℎ to both sides of this equation would give us four-thirds 𝑟 equals ℎ.

To find 𝑟, we divide both sides of the equation by four-thirds, which is equivalent to multiplying by three-quarters. Therefore, we have 𝑟 equals three-quarters ℎ. We’re left then with 𝑟 equals zero or 𝑟 equals three-quarters ℎ. And as 𝑟, the radius, cannot be equal to zero, then our answer for the formula for 𝑟 in terms of ℎ is 𝑟 equals three-quarters ℎ.

We’ll now we look at the second question to find the volume of the two solids. We’re told that the height of the cylinder needs to be 18 inches. So let’s work with the formula for the volume of a cylinder. We can then substitute the value ℎ equals 18, which would give us that the volume is equal to 𝜋𝑟 squared 18. However, as we don’t have a value for 𝑟, then we need to use the fact that 𝑟 equals three-quarters ℎ to help us.

When 𝑟 equals three-quarters ℎ and ℎ is equal to 18, then we have 𝑟 is equal to three-quarters of 18. Since three times 18 gives us 54, then we have 𝑟 equals 54 over four, which we can further simplify as 27 over two. And now, we can also substitute 𝑟 equals 27 over two into the formula above. So the volume of the cylinder is equal to 𝜋 times 27 over two squared times 18. Using a calculator, we can evaluate this as 10,305.994700101. As we’re asked to give our answer to two decimal places, we check the third decimal digit to see if it is five or more. As it is not, then we don’t round our value up. So we have 10305.99 cubic inches. As both volumes are the same, then this is our answer for the second question.

Therefore, our answer for the first question is 𝑟 equals three-quarters ℎ. And the answer for the second question for the volume is 10305.99 cubic inches.