1. Use the volume to find the radius:

$\displaystyle Volume=\frac{4}{3}\pi r^{3}$

$\displaystyle 36\pi = \frac{4}{3}\pi r^{3}$

$\displaystyle \frac{3}{4}(36\pi) = (\frac{4}{3}\pi r^{3})\frac{3}{4}$

$\displaystyle 27=r^{3}$

$\displaystyle r=3$

2. Use the radius to find the diameter:

$\displaystyle d=2r=2\cdot 3=6$

This question relies on knowledge of the formula for volume of a sphere, which is as follows: $\displaystyle V=\frac{4}{3}\pi r^{3}}$ 

In this equation, we have two variables, $\displaystyle V$ and $\displaystyle r$. Additionally, we know that $\displaystyle V=36\pi$ and $\displaystyle r$ is unknown. You can begin by rearranging the volume equation so it is solved for $\displaystyle r$, then plug in $\displaystyle V$ and solve for $\displaystyle r$:

Rearranged form:

 $\displaystyle r=\sqrt[3]{\frac{3}{4\pi}V}$

Plug in $\displaystyle 36\pi$ for V

$\displaystyle r=\sqrt[3]{\frac{3}{4\pi}36\pi}$

Simplify the part under the cubed root

1) Cancel the $\displaystyle \pi$'s since they are in the numerator and denominator.

2) Simplify the fraction and the $\displaystyle 36$:

 $\displaystyle 36*\frac{3}{4}=27$

Thus we are left with 

$\displaystyle r=\sqrt[3]{27}$

Then, either use your calculator and enter $\displaystyle 27^{\frac{1}{3}}$ Or recall that $\displaystyle 3^{3}=27$ in order to find that $\displaystyle r=3$.

We're almost there, but we need to go a step further. Dodge the trap answer "$\displaystyle 3$" and carry on. Read the question carefully to see that we need the diameter, not the radius.

$\displaystyle d=2r$

So

$\displaystyle d=2*3=6$

$\displaystyle 6$ is our final answer.

To answer this question, we must calculate the volume of the ball using the equation for the volume of a sphere. The equation for the volume of a sphere is four-thirds multiplied by pi, which is then multiplied by the radius cubed. The equation can be written like this:

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

We are given the diameter of the sphere in the problem, which is $\displaystyle 4\:in$. To get the radius from the diameter, we divide the diameter by $\displaystyle 2$. So, for this data:

$\displaystyle radius=\frac{diameter}{2}=\frac{4}{2}=2$

We can then plug our newly found radius of two into the equation to find the volume. For this data:

$\displaystyle Volume = \frac{4}{3}\pi r^{3}=\frac{4}{3}\pi \cdot (2)^{3} = \frac{4}{3}\pi\cdot 8$

We then multiply $\displaystyle \frac{4}{3}$ by $\displaystyle 8$.

$\displaystyle \frac{4}{3}\pi\cdot 8=\frac{32}{3}\pi$

We finally substitute 3.14 for pi and multiply again to get our answer.

$\displaystyle \frac{32}{3}\pi = 33.5$

The question asked us to round to the nearest whole cubic inch. To do this, we round a number up one place if the last digit is a 5, 6, 7, 8, or 9, and we round it down if the last digit is a 1, 2, 3, or 4. Therefore:

$\displaystyle 33.5\rightarrow34$

Therefore our answer is $\displaystyle 34\:in^3$.

The formula for the volume of a sphere is:

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

To figure out how far the sphere rolled, we need to know the circumference, so we must first figure out radius. Solve the formula for volume in terms of radius:

 $\displaystyle 1436 = \frac{4}{3}\pi r^3$

$\displaystyle \frac{1077}{\pi} =r^3$

$\displaystyle 342.99 = r^3$

$\displaystyle r \approx 6.999932$

Since the answer asks us to round to the nearest integer, we are safe to round $\displaystyle r$ to $\displaystyle 7$ at this point.

To find circumference, we now apply our circumference formula:

$\displaystyle C = 2r\pi = 14\pi$

If our boulder rolled $\displaystyle 355$ times, it covered that many times its own circumference.

$\displaystyle 355\cdot 14\pi = 4970\pi \approx 15606$

Thus, our boulder rolled for $\displaystyle 15606ft$

To solve, simply remember that diameter is twice the radius. Don't be fooled when the radius is an algebraic expression and incorporates the arbitrary constant $\displaystyle d$. Thus,

$\displaystyle \textup{diameter}=2r=2*2d=4d$