propane tanks\npeople who live in isolated or rural areas have their own tanks of natural gas to run…

Explanation:

Step1: Find the volume formula of the tank

The tank is composed of a cylinder and two hemispheres (which is equivalent to one - sphere). The volume of a cylinder is $V_{cylinder}=\pi r^{2}h$ and the volume of a sphere is $V_{sphere}=\frac{4}{3}\pi r^{3}$. So the volume of the tank $V = \pi r^{2}h+\frac{4}{3}\pi r^{3}$. Given $h = 10$ feet, then $V=\pi r^{2}(10)+\frac{4}{3}\pi r^{3}=\pi r^{2}(10 + \frac{4}{3}r)$.

Step2: Find the radius and volume of the standard tank

The diameter of the standard tank is 6 feet, so the radius $r_1=3$ feet. Substitute $r = 3$ into the volume formula: [ \begin{align*} V_1&=\pi\times3^{2}(10+\frac{4}{3}\times3)\ &=9\pi(10 + 4)\ &=9\pi\times14\ &=126\pi \end{align*} ]

Step3: Find the volume of the new tank

The company wants to double the capacity of the standard tank, so $V_2 = 2V_1=2\times126\pi = 252\pi$.

Step4: Set up the equation for the new - tank volume and solve for $r$

Set $V_2=\pi r^{2}(10+\frac{4}{3}r)$, so $252\pi=\pi r^{2}(10+\frac{4}{3}r)$. Divide both sides by $\pi$: $252=r^{2}(10+\frac{4}{3}r)=10r^{2}+\frac{4}{3}r^{3}$. Multiply through by 3 to get rid of the fraction: $756 = 30r^{2}+4r^{3}$, or $4r^{3}+30r^{2}-756 = 0$. Divide through by 2: $2r^{3}+15r^{2}-378 = 0$. We can try to find a root of the equation by trial - and - error. We check some simple values of $r$. When $r = 4$: [ \begin{align*} 2\times4^{3}+15\times4^{2}-378&=2\times64 + 15\times16-378\ &=128+240 - 378\ &=368 - 378\ &=- 10 \end{align*} ] When $r = 4.5$: [ \begin{align*} 2\times(4.5)^{3}+15\times(4.5)^{2}-378&=2\times91.125+15\times20.25 - 378\ &=182.25+303.75 - 378\ &=486 - 378\ &=108 \end{align*} ] Since the function $y = 2r^{3}+15r^{2}-378$ is continuous, we can use a numerical method (such as the bisection method) or a graphing calculator to find the root. Using a graphing calculator or software, we find that $r\approx4.24$ feet.

Answer:

The radius of the new tank is approximately $4.24$ feet.