Question

the volume of a cylinder is given by $v = \pi r^{2}h$, where $r$ is the radius of the cylinder and $h$ is the height. which expression represents the volume of this can?$\bigcirc$ $3\pi x^{2}+4\pi x + 16\piLXB0\bigcirc$ $3\pi x^{3}+32\pi$$\bigcirc$ $3\pi x^{3}+20\pi x^{2}+44\pi x + 32\pi$done

Explanation:

Step1: Identify given values

$r = x+2$, $h = 3x+8$, $V=\pi r^2 h$

Step2: Expand $r^2$ first

$(x+2)^2 = x^2 + 4x + 4$

Step3: Multiply by $h$

$(x^2 + 4x + 4)(3x+8) = 3x(x^2 + 4x + 4) + 8(x^2 + 4x + 4)$
$= 3x^3 + 12x^2 + 12x + 8x^2 + 32x + 32$

Step4: Combine like terms

$3x^3 + (12x^2+8x^2) + (12x+32x) + 32 = 3x^3 + 20x^2 + 44x + 32$

Step5: Multiply by $\pi$

$V = \pi(3x^3 + 20x^2 + 44x + 32)$

Answer:

$3\pi x^3 + 20\pi x^2 + 44\pi x + 32\pi$ (the fourth option)