The volume of a cone is given by the formula V = (πr²)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.

time in seconds = (πr²)/(3w)

In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.

(πr²)/(3w) * (1/60) = π(r²)(h)/(180w)

The basic form for the volume of a cone is:

V = (1/3)πr²h

For this simple problem, we merely need to plug in our values:

V = (1/3)π13² * 6 = 169 * 2π = 338π in³

There are two things to be careful with here.  First, we must solve for the radius of the base. Secondly, note that the height is given in feet, not inches. Notice that all the answers are in cubic inches. Therefore, it will be easiest to convert all of our units to inches.

First, solve for the radius, recalling that C = 2πr, or, for our values 77π = 2πr. Solving for r, we get r = 77/2 or r = 38.5.

The height, in inches, is 24.

The basic form for the volume of a cone is: V = (1 / 3)πr²h

For our values this would be:

V = (1/3)π * 38.5² * 24 = 8 * 1482.25π = 11,858π in³

The general formula is given by $\displaystyle V = 1/3Bh = 1/3\pi r^{2}h$, where $\displaystyle r$ = radius and $\displaystyle h$ = height.

The diameter is 6 cm, so the radius is 3 cm.

$\displaystyle V=\frac{\pi 3^2\times 5}{3}=15\pi$

First we will calculate the volume of the cone

$\displaystyle V=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi\cdot 4^{2}\cdot 18=96\pi\ m^{3}$

Next we will determine the time it will take to fill that volume

$\displaystyle 96\pi \ m^{3}\times \frac{25\ s}{3\ m^{3}}=800\pi \ s$

We will then convert that into minutes

$\displaystyle 800\pi \ s\times \frac{1\ minute}{60\ seconds}=13.\overline{3}\pi \ minutes\approx 41.9\ minutes\approx 41\ minutes\ 53\ seconds$

Write the formula to find the volume of a cone.

$\displaystyle V=\frac{1}{3}\pi r^2 h$

Substitute the known values and simplify.

$\displaystyle V=\frac{1}{3}\pi (10)^2 (90)= \frac{1}{3}\pi (100)(90)= \pi (100)(30)=3000\pi$

To solve, simply use the formula for the volume of a cone. Thus,

$\displaystyle V=\frac{1}{3}\pi{r^2}h=\frac{1}{3}\pi*{3^2}*5=\frac{1}{3}\pi*9*5=15\pi$

To remember the formula for volume of a cone, it helps to break it up into it's base and height. The base is a circle and the height is just h. Now, just multiplying those two together would give you the formula of a cylinder (see problem 3 in this set). So, our formula is going to have to be just a portion of that. Similarly to volume of a pyramid, that fraction is one third.

To solve, simply use the formula for the area of a cone. Thus,

$\displaystyle V=\frac{4}{3}\pi{r^2}h=\frac{4}{3}*\pi*4^2*3=4*\pi*16=64\pi$

The volume of a right circular cone with radius $\displaystyle r$ and height $\displaystyle h$ is given by:

$\displaystyle V = \pi r^2 \frac{h}{3}$

Since the height of this cone is equal to its radius, we can say:

$\displaystyle V = \pi r^2 \frac{r}{3} = \frac{1}{3}\pi r^3$

Now, we can substitute our given volume into the equation and solve for our radius.

$\displaystyle 9\pi = \frac{1}{3}\pi r^3$

$\displaystyle 27\pi = \pi r^3$

$\displaystyle 27 = r^3$

$\displaystyle r = 3$

The volume of a right circular cone $\displaystyle V$ can be calculated from its height $\displaystyle h$ and the radius $\displaystyle r$ of its base using the formula

$\displaystyle V = \frac{1}{3} \pi r^{2} h$.

We are given $\displaystyle r = 7$, but not $\displaystyle h$.

$\displaystyle h$, $\displaystyle r$, and the slant height $\displaystyle l$ of a right circular cone are related by the Pythagorean Theorem:

$\displaystyle h^{2}+ r^{2}= l^{2}$

Setting $\displaystyle l = 18$ and $\displaystyle r = 7$, substitute and solve for $\displaystyle h$:

$\displaystyle h^{2}+ 7^{2}= 18^{2}$

$\displaystyle h^{2}+49= 324$

$\displaystyle h^{2}+49 - 49 = 324 - 49$

$\displaystyle h^{2} =275$

Taking the square root of both sides and simplifying the radical:

$\displaystyle h =\sqrt{275} = \sqrt{25} \cdot \sqrt{11} = 5 \sqrt{11}$

Now, substitute for $\displaystyle r$ and $\displaystyle h$ and evaluate:

$\displaystyle V = \frac{1}{3} \pi r^{2} h$

$\displaystyle V = \frac{1}{3} \pi \cdot 7^{2} \cdot 5 \sqrt{11}$

$\displaystyle V = \frac{1}{3} \pi \cdot 49 \cdot 5 \sqrt{11}$

$\displaystyle V = \frac{245}{3} \pi \sqrt{11}$