Video: Finding Volume of Composite Solids in a Real-World Context

A solid cone with a radius of 5 inches and a height of 20 inches is placed into a cylindrical tank full of water with the same height and radius. How much water is displaced by the cone? Give your answer in cubic inches to two decimal places. How much water is left in the cylindrical tank? Give your answer in cubic inches accurate to two decimal places.

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Video Transcript

A solid cone with a radius of five inches and a height of 20 inches is placed into a cylindrical tank full of water with the same height and radius. How much water is displaced by the cone? Give your answer in cubic inches to two decimal places. How much water is left in the cylindrical tank? Give your answer in cubic inches accurate to two decimal places.

So let’s begin by having a look at the first question. To start, we can think about how we could represent this in a sketch. We know that we have a cylindrical tank and into it we put a cone with a radius of five inches and a height of 20 inches. So here, we have our cylinder. And since the height and the radius of the cone are the same as the cylinder, this means that there will be no extra space around the flat face of our cone. When we draw the cone within the cylinder, it doesn’t matter whether we have the flat face of the cone facing downwards or upwards. In the first question, we’re asked how much water is displaced by the cone. And the water that will be displaced is equal to the amount of space that the cone takes up, in other words, the volume. To find the volume, we use the formula the volume of a cone equals 𝜋𝑟 squared times ℎ over three, where ℎ is the height.

So let’s begin by writing the formula and substituting in our values. Since we know that our radius 𝑟 is five and the height is 20, this will give us the volume equals 𝜋 times five squared times 20 over three. When we have values put next to each other like this, it’s the same as saying 𝜋 times five squared times 20 over three. And it doesn’t matter which order we perform the multiplication in. And since five squared is five times five, we can simplify this as 25𝜋 times 20 over three. We could further simplify this as 500 over three times 𝜋. At this point, since we’re asked for an answer in two decimal places, this means that we need to evaluate 503 over 𝜋. So we’re going to need to use a calculator which will give us 523.5987756.

To round our answer to two decimal places, we need to check the third decimal place digit. If it’s five or more, that means we round our second decimal digits up by one. As our digit, eight, is five or more, this means that our digit, nine, will round up to zero. And in turn, our next digit on the left must also round up to six. And so our answer for the water that’s displaced by the cone is the same as our volume. And it’s 523.60 cubic inches. So now, let’s have a look at the second question that we were asked.

How much water is left in the cylindrical tank? Give your answer in cubic inches accurate to two decimal places. So whenever the cone was put into the cylindrical tank, the water flowed out and the water that flowed out was equal to the volume of the cone. This means that the water that’s left in the tank is the water surrounding the cone. And to figure out exactly how much is left, we need to know how much was in the tank at the very start. So we need to work out the volume of the cylinder. And we can do this using the formula, the volume of a cylinder equals 𝜋𝑟 squared ℎ, where 𝑟 is the radius and ℎ is the height. So we write our formula and substitute in our values. The radius 𝑟 is five and the height ℎ is 20, giving us the volume equals 𝜋 times five squared times 20.

Simplifying this will give us 𝜋 times 25 times 20 which is equal to 500𝜋. As a decimal value, this is equal to a 1570.796327 cubic inches. And we’re not going to round our answer just yet. So since in the question we were asked how much water is left in the tank, we’re not quite finished with our answer just yet. To find the water that’s left, we need to calculate the volume of the cylinder take away the volume of the cone, which is 1570.796327 subtract 523.5987756. We used the unrounded answer for the volume of our cone rather than the rounded answer, as this will make our final answer more accurate.

And simplifying this will give us 1047.197551. As we need to round our answer to two decimal places, this means that we need to check the digit in the third decimal place column. Since the digit seven is equal to five or more, this means we must run our digit nine up. And since the nine becomes a zero, this means we round our next column on the left up by one. And so our final answer for the water that’s left in the tank is 1047.20 cubic inches.

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