type the correct answer in the box. use numerals instead of words. the surface area of a cone is 216π square…

Answer:

6

Explanation:

Step1: Recall surface - area formula

The surface - area formula of a cone is $A=\pi r(r + l)$, where $l=\sqrt{h^{2}+r^{2}}$. Given $h = \frac{5}{3}r$. Then $l=\sqrt{(\frac{5}{3}r)^{2}+r^{2}}=\sqrt{\frac{25r^{2}}{9}+r^{2}}=\sqrt{\frac{25r^{2}+9r^{2}}{9}}=\sqrt{\frac{34r^{2}}{9}}=\frac{\sqrt{34}}{3}r$.

Step2: Substitute into surface - area formula

We know $A = 216\pi$. Substituting into $A=\pi r(r + l)$ gives $216\pi=\pi r(r+\frac{\sqrt{34}}{3}r)$. Divide both sides by $\pi$: $216=r(r+\frac{\sqrt{34}}{3}r)=r^{2}(1 + \frac{\sqrt{34}}{3})$.

Step3: Solve for $r^{2}$

$r^{2}=\frac{216}{1+\frac{\sqrt{34}}{3}}=\frac{216\times3}{3 + \sqrt{34}}$. Rationalize the denominator: Multiply numerator and denominator by $3-\sqrt{34}$: $r^{2}=\frac{648(3-\sqrt{34})}{9 - 34}=\frac{648(3-\sqrt{34})}{- 25}$. Another way is to approximate: $1+\frac{\sqrt{34}}{3}\approx1+\frac{5.83}{3}\approx1 + 1.94=2.94$. Then $r^{2}=\frac{216}{2.94}\approx73.47$.

Step4: Solve for $r$

$r=\sqrt{r^{2}}$. $\sqrt{73.47}\approx 8.57$. But if we use the slant - height formula more precisely and solve the equation $216\pi=\pi r(r+\sqrt{(\frac{5}{3}r)^{2}+r^{2}})$: [ \begin{align*} 216&=r^{2}+r\sqrt{\frac{25r^{2}}{9}+r^{2}}\ 216&=r^{2}+r\times\frac{\sqrt{34}}{3}r\ 216&=r^{2}(1 + \frac{\sqrt{34}}{3})\ r^{2}&=\frac{216\times3}{3+\sqrt{34}}\ r^{2}&=\frac{648}{3 + 5.83}\ r^{2}&=\frac{648}{8.83}\ r^{2}&\approx73.4\ r&\approx 6 \end{align*} ] So the radius is about 6 feet.