the volume of a cylinder is given by the formula $v = \\pi r^{2}h$, where $r$ is the radius of the cylinder…

the volume of a cylinder is given by the formula $v = \\pi r^{2}h$, where $r$ is the radius of the cylinder and $h$ is the height. suppose a cylindrical can has radius $(x + 8)$ and height $(2x + 3)$. which expression represents the volume of the can?\n$\\pi x^{3}+19\\pi x^{2}+112\\pi x + 192\\pi$\n$2\\pi x^{3}+35\\pi x^{2}+80\\pi x + 48\\pi$\n$2\\pi x^{3}+35\\pi x^{2}+176\\pi x + 192\\pi$\n$4\\pi x^{3}+44\\pi x^{2}+105\\pi x + 72\\pi$

the volume of a cylinder is given by the formula $v = \\pi r^{2}h$, where $r$ is the radius of the cylinder and $h$ is the height. suppose a cylindrical can has radius $(x + 8)$ and height $(2x + 3)$. which expression represents the volume of the can?\n$\\pi x^{3}+19\\pi x^{2}+112\\pi x + 192\\pi$\n$2\\pi x^{3}+35\\pi x^{2}+80\\pi x + 48\\pi$\n$2\\pi x^{3}+35\\pi x^{2}+176\\pi x + 192\\pi$\n$4\\pi x^{3}+44\\pi x^{2}+105\\pi x + 72\\pi$

Answer

Explanation:

Step1: Substitute radius and height into formula

Given $r = x + 8$ and $h=2x + 3$, and $V=\pi r^{2}h$. First find $r^{2}=(x + 8)^{2}=x^{2}+16x + 64$ using the formula $(a + b)^{2}=a^{2}+2ab + b^{2}$ with $a=x$ and $b = 8$.

Step2: Multiply $r^{2}$ by $h$ and $\pi$

$V=\pi(x^{2}+16x + 64)(2x + 3)$. Expand $(x^{2}+16x + 64)(2x + 3)$: [ \begin{align*} &(x^{2}+16x + 64)(2x + 3)\ =&x^{2}(2x+3)+16x(2x + 3)+64(2x + 3)\ =&2x^{3}+3x^{2}+32x^{2}+48x+128x + 192\ =&2x^{3}+35x^{2}+176x + 192 \end{align*} ] Then multiply by $\pi$ to get $V = 2\pi x^{3}+35\pi x^{2}+176\pi x+192\pi$.

Answer:

$2\pi x^{3}+35\pi x^{2}+176\pi x + 192\pi$