Question

problem 4 applying multiplication of binomials got it? what is the total surface area of a cylinder with radius ( x + 2 ) and height ( x + 4 )? write your answer as a polynomial in standard form.

Explanation:

Step1: Recall cylinder surface area formula

The total surface area of a cylinder is $SA = 2\pi r^2 + 2\pi rh$, where $r$ is radius and $h$ is height.

Step2: Substitute given $r$ and $h$

Substitute $r = x+2$, $h = x+4$:
$SA = 2\pi (x+2)^2 + 2\pi (x+2)(x+4)$

Step3: Expand $(x+2)^2$

$(x+2)^2 = x^2 + 4x + 4$, so:
$SA = 2\pi(x^2 + 4x + 4) + 2\pi(x+2)(x+4)$

Step4: Expand $(x+2)(x+4)$

$(x+2)(x+4) = x^2 + 6x + 8$, so:
$SA = 2\pi(x^2 + 4x + 4) + 2\pi(x^2 + 6x + 8)$

Step5: Distribute $2\pi$ to each term

$SA = 2\pi x^2 + 8\pi x + 8\pi + 2\pi x^2 + 12\pi x + 16\pi$

Step6: Combine like terms

Combine $x^2$ terms: $2\pi x^2 + 2\pi x^2 = 4\pi x^2$
Combine $x$ terms: $8\pi x + 12\pi x = 20\pi x$
Combine constant terms: $8\pi + 16\pi = 24\pi$
$SA = 4\pi x^2 + 20\pi x + 24\pi$

Answer:

$4\pi x^2 + 20\pi x + 24\pi$