Math Problem Solver

The volume of a cone is $\frac{29070312500 \pi}{3} \approx 30442360062.52\, \text{mm}^3$.

Note: Approximate solutions are rounded to the nearest hundredths place.

Given: $$\begin{aligned}d&=15250\\h&=500\end{aligned}$$

Volume

The volume of a cone is given by the equation: $$\begin{aligned}V&=\frac{1}{3} \pi r^{2} h\end{aligned}$$ where $V$ is volume, $r$ is radius, and $h$ is height.

Need:$$\begin{aligned}d&=2 r\end{aligned} \quad \Rightarrow \quad \begin{aligned}15250&=2 r\end{aligned} \quad \Rightarrow \quad \begin{aligned}r&=7625\end{aligned}$$

Solution:$$\begin{aligned}V&=\frac{1}{3} \pi r^{2} h\\V&=\frac{1}{3} \pi \left(7625\right)^{2} \left(500\right)\\V&=\frac{29070312500 \pi}{3}\end{aligned}$$

A right cone is a three dimensional solid with one vertex, one flat face, and one curved surface pointed towards the top, called apex. The circular face is called the base. The radius r is the distance between the center of the base and any point on the circumference of the base. The diameter d is twice the radius, forming a line segment passing through the center of the base. The height h is the distance between the center of the base and the apex, forming a line segment perpendicular to the base. The slant height H is the distance between the apex and any point on the circumference of the base.

The slant height H of a cone is the hypotenuse of the right triangle with legs being the radius and height. $$H = \sqrt{r^{2} + h^{2}}$$

The (base) circumference C, or base perimeter P_B, of a cone is the distance around the base.$$\begin{aligned} C = P_{B} &= 2 \pi r \\ &= \pi d \end{aligned}$$

The base area A_B of a cone is the region enclosed inside the base. $$A_{B} = \pi r^{2}$$

The lateral surface area A_L of a cone is the area of the curved surface. $$\begin{aligned} A_{L} &= \pi r H \\ &= \pi r \sqrt{r^{2} + h^{2}} \end{aligned}$$

The surface area S of a cone is the region occupied by its base and curved surface. $$\begin{aligned} S &= A_{B} + A_{L} \\ &= \pi r (r + H) \\ &= \pi r (r + \sqrt{r^{2} + h^{2}}) \end{aligned}$$

The volume V of a cone is the amount of space it occupies. $$\begin{aligned} V &= \frac{1}{3} A_{B} h \\ &= \frac{1}{3} \pi r^{2} h \end{aligned}$$