You are 3D printing a ball which will be an integral piece to the project you are creating. If you need the ball to have a radius of 12cm, what volume of material will be required?

We are asked to find the volume of a sphere, to do so, we should use the formula for volume of a sphere.

\(\displaystyle V_{sphere}=\frac{4}{3} \pi r^3\)

So, let's plug in our radius and solve:

\(\displaystyle V_{sphere}=\frac{4}{3} \pi (12cm)^3=\frac{4}{3} \pi 1728cm^3=7234.56 cm^3\)

So our answer is:

\(\displaystyle 7234.56 cm^3\)

You are given a spherical floating speaker for your birthday. If the speaker has a radius of 5 inches, what is its volume?

We need to find the volume of a sphere. To do so, we need the following formula:

\(\displaystyle V=\frac{4}{3}\pi r^3\)

We know r is 5, so just plug and simplify

\(\displaystyle V=\frac{4}{3}\pi (5in)^3\approx 523.56in^3\)

The formula for the surface area \(\displaystyle A\) of a sphere, given its radius \(\displaystyle r\), is

\(\displaystyle A= 4 \pi r^{2}\).

Set \(\displaystyle A = 20 \pi\) and solve for \(\displaystyle r\) to get the radius.

\(\displaystyle 4 \pi r^{2} = 20 \pi\)

Divide both sides by \(\displaystyle 4 \pi\):

\(\displaystyle \frac{4 \pi r^{2}}{4 \pi} = \frac{20 \pi}{4 \pi}\)

\(\displaystyle r^{2} = 5\)

Take the positive square root of both sides to obtain the radius:

\(\displaystyle r = \sqrt{5}\)

The formula for the volume \(\displaystyle V\) of a sphere, given its radius \(\displaystyle r\), is

\(\displaystyle V = \frac{4 \pi r^{3}}{3}\)

Substitute \(\displaystyle \sqrt{5}\) for \(\displaystyle r\) in this formula and evaluate the expression:

\(\displaystyle V = \frac{4 \pi (\sqrt{5})^{3}}{3}\)

\(\displaystyle = \frac{4 \pi \cdot \sqrt{5}\cdot \sqrt{5} \cdot \sqrt{5}}{3}\)

\(\displaystyle = \frac{4 \pi \cdot5 \cdot \sqrt{5}}{3}\)

\(\displaystyle = \frac{20 \pi \sqrt{5}}{3}\)

Recall how to find the volume of a sphere:

\(\displaystyle \text{Volume}=\frac{4}{3}\pi r^3\)

Plug in the given radius to solve for the volume.

\(\displaystyle \text{Volume}=\frac{4}{3}\pi (8^3)=\frac{2048\pi}{3}=2144.66\)

Make sure to round to two places after the decimal point.

The formula for the volume of a sphere is \(\displaystyle (4/3)\Pi r^3\), with \(\displaystyle r\) standing for radius. Since you know that the radius of the sphere is \(\displaystyle 3\), all you have to do is plug in \(\displaystyle 3\) for \(\displaystyle r\) and solve. Once you do that you learn that the volume is \(\displaystyle 36\Pi\).

Start by finding the volume of one water balloon.

Recall how to find the volume of a sphere:

\(\displaystyle \text{Volume}=\frac{4}{3}\pi r^3\)

Plug in the given value for the radius.

\(\displaystyle \text{Volume}=\frac{4}{3}\pi (3.25)^3=143.79\)

Now, since one water balloon will require \(\displaystyle 143.79 in^3\) of water, divide the total volume of water by this value to find how many balloons can be filled.

\(\displaystyle \text{Number of balloons filled}=\frac{5000}{143.79}=34.77\)

Since the question asks for the number of complete balloons that can be filled, we will have to round down to the nearest whole number, \(\displaystyle 34\).

Consider a tube which is 3 ft wide and 18 ft long.

Find the volume of the largest sphere which could fit within the tube described above. 

To find the volume of a sphere, we simply need its radius

\(\displaystyle V=\frac{4}{3}\pi r^3\)

Now, the largest sphere which will fit within the tube will need to have a radius equal to the tube. Therefore, we can say our radius must be half the diameter, making it 1.5 ft.

Next, plug 1.5 ft into our formula to find our Volume

\(\displaystyle V=\frac{4}{3}\pi (1.5ft)^3=4.5\pi ft^3\)

So, our answer is:

\(\displaystyle V= 4.5\pi ft^3\)

This problem is deceptively simple. In order to solve for the volume of a sphere, all you need is the formula: \(\displaystyle V=\frac{4}{3}\pi r^3\), where r is the radius. 

This problem has provided additional information alongside the pertinent information. We don't need to know what the circumference is if we have been provided with the radius. 

Therefore, this problem can be quickly solved for by substituting \(\displaystyle x\) for \(\displaystyle r\) in the volume formula. 

\(\displaystyle V=\frac{4}{3}\pi x^3\)

therefore, the volume of this sphere is \(\displaystyle \frac{4}{3}\pi x^3\)

This problem is deceptively simple. In order to solve for the volume of a sphere, all you need is the formula: \(\displaystyle V=\frac{4}{3}\pi r^3\), where r is the radius. 

This problem has provided us with the diameter, so we just need to do a little bit of work to solve for the radius. What is the relationship between radius and diameter? The diameter is twice the radius. Or in math speak: \(\displaystyle d = 2r\). This means we can solve for our radius by taking half of the diameter. Therefore, the radius will be \(\displaystyle 18.\) 

Now that we have r, we can substitute in the value for r and solve for the volume!

\(\displaystyle V=\frac{4}{3}\pi 18^3\)

\(\displaystyle V=\frac{4}{3}\pi \times 5832\)

\(\displaystyle V=\frac{5832 \times 4}{3}\pi\)

\(\displaystyle V=7776\pi\)

This problem is very easy to solve as long as you have the volume formula for a sphere handy. 

The formula is \(\displaystyle V=\frac{4}{3}\pi r^3\), where r is the radius. 

The problem provides us with V, the volume. If we substitute in the volume, the only unknown in the problem is r. This is exactly what we want. 

\(\displaystyle 284\pi=\frac{4}{3}\pi r^3\)

The goal is to get r by itself. We can begin this process by dividing by \(\displaystyle \pi\).

\(\displaystyle \frac{284\pi}{{\color{Red} \pi}}=\frac{\frac{4}{3}{\color{Red} \pi} r^3}{{\color{Red} \pi}}\)

\(\displaystyle 284=\frac{4}{3} r^3\)
Now we may multiply both sides of the equation by \(\displaystyle \frac{3}{4}\) to remove the fraction from the right side of the equals sign. This allows us to get closer to solving for r. 

\(\displaystyle 284 \times \frac{3}{4}=\frac{4}{3} r^3 \times \frac{3}{4}\)

\(\displaystyle \frac{284 \times 3}{4}= r^3\)

\(\displaystyle 213= r^3\)

Now we need to take the cubed root of each side and we will have solved for the radius!

\(\displaystyle 5.97= r\)