Recall that the surface area of a sphere is found by the equation:

\(\displaystyle SA = 4\pi r^2\)

For our data, this means:

\(\displaystyle 169\pi = 4\pi r^2\)

Solve for \(\displaystyle r\). First, divide by \(\displaystyle 4\pi\):

\(\displaystyle 42.25=r^2\)

Take the square root of both sides:

\(\displaystyle r = 6.5\)

Given the volume of the sphere, \(\displaystyle 36\prod \textup{units}^{3}\), you need to use the formula for volume of a sphere \(\displaystyle \left ( \left ( \frac{4}{3} \right )\cdot \prod \cdot r^{3}\right )\) and work backwards to find the radius. I would multiply both sides by \(\displaystyle \frac{3}{4}\) to get rid of the \(\displaystyle \frac{4}{3}\) in the formula. You then have \(\displaystyle 27\prod=\prod r^{3}\). Next, divide both sides by \(\displaystyle \prod\) so that all vyou have left is \(\displaystyle 27=r^{3}\). Finally take the cube root of \(\displaystyle 27\), to get \(\displaystyle 3\) units for the radius.

Solving this problem requires recognizing that since the cube is circumscribed by the sphere, both solids share the same center. Now it is just a matter of finding the diagonal of the cube, which will double as the diameter of the sphere (by definition, any straight line which passes through the center of the sphere). The formula for the diagonal of a cube is \(\displaystyle D = s\sqrt{3}\), where \(\displaystyle s\) is the length of the side of a cube. (This occurs because you must use the Pythagorean theorem once for each 2-dimensional "corner" you travel to find the diagonal for a 3-dimensional shape, but for the ACT it's much faster to memorize the formula.)

In this case:

\(\displaystyle D = s\sqrt3 = 25\sqrt3\)

Since the radius is half the diameter, divide the result in half:

\(\displaystyle r = \frac{D}{2} = \frac{25\sqrt{3}}{2}\)

\(\displaystyle r = \frac{25\sqrt{3}}{2}\)

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,

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

Calculate the slant height height of the cone using the Pythagorean Theorem. The height will be the height of the cone, the base will be the radius, and the hypotenuse will be the slant height.

\(\displaystyle s^2=r^2+h^2\)

\(\displaystyle s^2=(5)^2+(12)^2=25+144=169\)

\(\displaystyle s=\sqrt{169}=13\)

The surface area of the cone (excluding the base) is given by the formula \(\displaystyle A=\pi rs\). Plug in our values to solve.

\(\displaystyle A_{cone}=\pi(5)(13)=65\pi\ cm^2\)

The surface area of a sphere is given by \(\displaystyle A=4\pi r^2\) but we only need half of the sphere, so the area of a hemisphere is \(\displaystyle A=2\pi r^2\).

\(\displaystyle A_{hemisphere}=2\pi(5)^2=50\pi\ cm^2\)

So the total surface area of the composite figure is \(\displaystyle 65\pi\ cm^2+50\pi\ cm^2=115\pi\ cm^2\).

The first step is to use the surface area formula to find the radius of the sphere.

\(\displaystyle A=4\pi r^2\)

\(\displaystyle 16\pi=4\pi r^2\)

\(\displaystyle 16=4r^2\)

\(\displaystyle 4=r^2\)

\(\displaystyle 2=r\)

The next step is to plug the value of the radius into the volume formula.

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

\(\displaystyle V=\frac{4}{3}\pi (2)^3\)

\(\displaystyle V=\frac{4}{3}\pi(8)\)

\(\displaystyle V=\frac{32}{3}\pi cm^3\)