13. Volumes of 3D Shapes

Question 1

Find the volume of the following hexagonal prism.

Select the correct answer from the list below:

A: 284 \text{cm}^3

B: 382\text{cm}^3

C: 444 \text{cm}^3

D: 528 \text{cm}^3

CORRECT ANSWER:     C: 444 \text{cm}^3

WORKED SOLUTION:

The volume of any prism can be calculated from multiplying the area of the face of that prism by the length.

Volume = 12\times 37 = 444\text{cm}^3

Question 2

Find the volume of the following triangular prism with a height of 10cm

Select the correct answer from the list below:

A: 150\text{cm}^3

B: 300\text{cm}^3

C: 12\text{cm}^3

D: 245\text{cm}^3

CORRECT ANSWER:  A: 150\text{cm}^3

WORKED SOLUTION:

To find the volume of a prism, we just need to multiply the length by the cross-sectional area. However, we don’t have the area here so we need to find that first.

\text{Triangle Area }=\frac{1}{2}\times\text{base}\times\text{height} \text{Triangle Area }=\frac{1}{2}\times6\times10=30\text{cm}^2

Now that we have the cross-section’s area, we just need to multiply it with the length

Question 3

Find the height of the cube with volume 250m^3. Give your answer to 3 significant figures.

Select the correct answer from the list below:

A: 6.3m

B: 7.29m

C: 8.33m

D: 9.1m

CORRECT ANSWER:     A: 6.3m

WORKED SOLUTION:

If we think about a cube, we know that all the sides are the same length

And how do we find the volume?… we multiply them all together.

But we know the volume of this cube is 250m^3, so we can set them equal.

So, to find just x, we need to do the opposite of cubing, which is cuberooting

Question 4

Below is the diagram for the cap of a tube of toothpaste.

The cap begins as an empty cylinder before a solid cone with same radius and height of the cap is put at the bottom.

Find the amount of empty space inside the cylinder.

Give your answer to 3 significant figures.

Select the correct answer from the list below:

A: 83.8mm^3

B: 103mm^3

C: 41.9mm^3

D: 27.6mm^3

CORRECT ANSWER:    C: 41.9mm^3

WORKED SOLUTION:

To find the amount of empty space we need to find the volume of the volume of the cylinder (the total amount of space inside) and then subtract the volume of the cone (space taken up). This will give us the amount of empty space that is left.

\text{Cylinder Volume }=\pi\times \text{radius}^2\times\text{height} \text{Cylinder Volume }=\pi\times 2^2\times5=20\pi\text{mm}^3

\text{Volume of Cone }=\frac{1}{3}\times\pi\times\text{radius}^2\times\text{height}
\text{Volume of Cone }=\frac{1}{3}\times\pi\times2^2\times5=\frac{20}{3}\pi

Now, all we need to do is subtract them

\text{Empty Space }=\text{ Cylinder Volume } - \text{ Volume of Cone }
\text{Empty Space }=20\pi-\frac{20}{3}\pi=41.9\text{mm}^3

Question 5

Below is the diagram for a silo with a volume of 1600m^3.

The silo is made up of a cylinder and a hemisphere, both with radius 5m. Determine the overall height of the silo.

Select the correct answer from the list below:

A: 17m

B: 25m

C: 12m

D: 22m

CORRECT ANSWER:    D: 22m

WORKED SOLUTION:

To find the height of the silo we need to know the height of the hemisphere and the height of the cylinder, then put them together. The height of the hemisphere is pretty easy, as it will just be the radius, 5m. The height of the cylinder is a little trickier, as we need to know the volume first. Let’s break this up into two pieces.

Hemisphere

\text{Volume of Hemisphere}=\frac{1}{2}\times\text{Volume of Sphere}
\text{Volume of Hemisphere}=\frac{1}{2}\times\frac{4}{3}\times\pi\times \text{radius}3
\text{Volume of Hemisphere}= \frac{2}{3}\times\pi\times\text{radius}^3
\text{Volume of Hemisphere}= \frac{2}{3}\times\pi \times 5^3 =\frac{250}{3}\pi
Cylinder

\text{Volume of Cylinder}=\text{Total Volume} - \text{Volume of Hemisphere}
\text{Volume of Cylinder} =1600-\frac{250}{3}\pi=1338.2\text{m}^3

Now we need to link the height of the cylinder to the volume, which we can do with the formula.

\text{Volume of Cylinder} =\pi \times \text{radius}^2\times\text{height}
\text{Volume of Cylinder} =\pi \times 5^2\times\text{height}
\text{Volume of Cylinder} =25\pi\times\text{height}

Now, we can set our two equations for the volume of a cylinder equal to each other

\text{Volume of Cylinder} =1338.2\text{m}^3
\text{Volume of Cylinder} =25\pi\times\text{height}
25\pi\times\text{height}=1338.2

So, we need to divide both sides by 25\pi to get “height” by itself

\text{height}=1338.2\div(25\pi) \text{height}=17\text{m}

Now, to find the overall height we need to add the height of the cylinder and the height of the hemisphere
\text{Total Height }=17+5=22\text{m}