Question

for the right circular cone shown, b is a point on the circumference of the base, and the length of segment ab (not shown) is 86 centimeters. if the height of the cone is 43 centimeters and the volume of the cone is kπ cubic centimeters, what is the value of k? note: figure not drawn to scale.

Explanation:

Step1: Find the radius from the circumference

The formula for the circumference of a circle is $C = 2\pi r$. Given $C=86$, then $2\pi r=86$, so $r=\frac{86}{2\pi}=\frac{43}{\pi}$.

Step2: Use the volume formula for a cone

The volume formula of a cone is $V=\frac{1}{3}\pi r^{2}h$. Substitute $r = \frac{43}{\pi}$ and $h = 43$ into the formula:
\[

$$\begin{align*} V&=\frac{1}{3}\pi(\frac{43}{\pi})^{2}\times43\\ &=\frac{1}{3}\pi\times\frac{43^{2}}{\pi^{2}}\times43\\ &=\frac{43^{3}}{3\pi} \end{align*}$$

\]
Since $V = k\pi$, then $\frac{43^{3}}{3\pi}=k\pi$. Cross - multiply to get $43^{3}=3k\pi^{2}$, and $k=\frac{43^{3}}{3\pi^{2}}$. But if we assume the intended formula is $V=\frac{1}{3}\pi r^{2}h=k$, substituting $r=\frac{43}{\pi}$ and $h = 43$:
\[

$$\begin{align*} V&=\frac{1}{3}\pi\times\frac{43^{2}}{\pi^{2}}\times43\\ &=\frac{43^{3}}{3\pi} \end{align*}$$

\]
We know $V = k\pi$, so $\frac{43^{3}}{3\pi}=k\pi$, then $k=\frac{43^{3}}{3\pi^{2}}$. If we work in terms of the volume formula $V=\frac{1}{3}\pi r^{2}h$ and equate it to $k\pi$ directly:
\[

$$\begin{align*} \frac{1}{3}\pi r^{2}h&=k\pi\\ \frac{1}{3} r^{2}h&=k \end{align*}$$

\]
Substitute $r=\frac{86}{2\pi}=\frac{43}{\pi}$ and $h = 43$:
\[

$$\begin{align*} k&=\frac{1}{3}\times(\frac{43}{\pi})^{2}\times43\\ &=\frac{43^{3}}{3\pi^{2}} \end{align*}$$

\]
\[

$$\begin{align*} k&=\frac{79507}{3\pi^{2}}\\ &\approx\frac{79507}{3\times9.8696}\\ &\approx\frac{79507}{29.6088}\\ &\approx2685.19 \end{align*}$$

\]
Another way:
The volume formula of a cone $V=\frac{1}{3}\pi r^{2}h$. Given $C = 2\pi r=86$, so $r=\frac{43}{\pi}$ and $h = 43$.
\[

$$\begin{align*} V&=\frac{1}{3}\pi\times(\frac{43}{\pi})^{2}\times43\\ &=\frac{43^{3}}{3\pi} \end{align*}$$

\]
Since $V = k\pi$, we have $\frac{43^{3}}{3\pi}=k\pi$. Solving for $k$ gives $k=\frac{43^{3}}{3\pi^{2}}$.
\[

$$\begin{align*} k&=\frac{79507}{3\pi^{2}} \end{align*}$$

\]
If we use the correct relationship from $V=\frac{1}{3}\pi r^{2}h=k\pi$:
\[

$$\begin{align*} \frac{1}{3}r^{2}h&=k\\ \frac{1}{3}\times(\frac{43}{\pi})^{2}\times43&=k\\ k&=\frac{43^{3}}{3\pi^{2}} \end{align*}$$

\]
\[

$$\begin{align*} k&=\frac{79507}{3\pi^{2}}\approx\frac{79507}{3\times 9.8696}\approx2685.19 \end{align*}$$

\]
The volume formula of a cone $V=\frac{1}{3}\pi r^{2}h$.
Since $C = 2\pi r = 86$, $r=\frac{43}{\pi}$ and $h = 43$.
\[

$$\begin{align*} V&=\frac{1}{3}\pi\times(\frac{43}{\pi})^{2}\times43\\ &=\frac{43^{3}}{3\pi} \end{align*}$$

\]
As $V=k\pi$, we get $k = \frac{43^{3}}{3\pi^{2}}=\frac{79507}{3\pi^{2}}\approx2685.19$

Answer:

$\frac{43^{3}}{3\pi^{2}}$