Video: US-SAT04S4-Q18-903102405048

The piston bowl shown in the figure consists of a cylinder with a hemispherical cavity at the top. Which of the following is closest to the volume of metal required to construct this piston bowl? [A] 2.73 m³ [B] 1.69 m³ [C] 2.47 m³ [D] 1.95 m³

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Video Transcript

The piston bowl shown in the figure consists of a cylinder with a hemispherical cavity at the top. Which of the following is closest to the volume of metal required to construct this piston bowl? A) 2.73 meters cubed. B) 1.69 meters cubed. C) 2.47 meters cubed. Or D) 1.95 meters cubed.

To calculate the volume of this shape, we’ll need to do two things. First, we’ll need to know the volume of the cylinder without the hemispherical cavity at the top. To calculate that, we use volume equals 𝜋𝑟 squared times the height, where 𝑟 is the radius of the base circle. And ℎ is the height, the distance from one base to the other in the cylinder.

After that, we need to find the volume of this hemispherical cavity. Based on its description, we know that a hemisphere is half of a sphere. If the volume of a sphere is four-thirds 𝜋𝑟 cubed, then the volume of a hemisphere would be equal to one-half four-thirds 𝜋𝑟 cubed. And when we multiply one-half by four-thirds, four over two reduces to two. And that means the simplified form of this formula would be the volume equals two-thirds 𝜋𝑟 cubed, where 𝑟 is the radius of the base.

So, let’s start with our cylinder. The volume equals 𝜋𝑟 squared times the height. The first thing we’ll look for is the radius of the base. We’re not given a radius here. We’re given a diameter. We know the distance all the way across the circle is 1.5 meters. If we take that and divide it by two, we’ll find the radius. So, we can say that the radius of our cylinder is 0.75. It’s half of 1.5.

We plug in 0.75 for our radius. And then, we recognize that our height is 1.25 meters, so we plug that in for ℎ. We could use a calculator to calculate 𝜋 times 0.75 squared times 1.25. When we do that, we get to 2.20893 continuing. And we don’t want to round at this stage, so we’ll leave it like this. And this is a measurement of volume, so that is meters cubed.

Now, we’ll consider the volume of our hemisphere. We need to be really careful here because the formula for our hemisphere involves a radius and the formula for our cylinder involves a radius. But if you look at the figure carefully, the radius of the cylinder is larger than the radius of the hemisphere. The space between these two is 0.25 meters.

And that means the radius of the hemisphere is 0.25 meters less than the radius of the cylinder. Since the radius of the cylinder was 0.75, the radius of the hemisphere will be 0.5 meters. And that’s what we need to plug into this formula. Two-thirds times 𝜋 times 0.5 cubed. If we plug this value into our calculator, we get 0.26179 continuing. And since this is a measure of volume, it’s meters cubed.

To find the volume of this piston bowl, we now need to take the volume of the cylinder and subtract the space that the hemisphere takes up, 2.20893 minus 0.26179. Just to know about these two values, you can keep them stored in your calculator and then subtract them to get the most accurate answer. And then, we get 1.94713 continuing.

Looking at our answer choices, they’re all given to two decimal places. If we round this value to two decimal places, there’s a seven to the right, and that means we need to round up to 1.95. And since it’s volume, it’s a measure of meters cubed, which is option D.

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