Question

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?

\\(\pi x^3 + 19\pi x^2 + 112\pi x + 192\pi\\)
\\(2\pi x^3 + 35\pi x^2 + 80\pi x + 48\pi\\)
\\(2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi\\)
\\(4\pi x^3 + 44\pi x^2 + 105\pi x + 72\pi\\)

Explanation:

Identify given values and formula

We are given the volume formula:

$$V = \pi r^2 h$$

The radius is:

$$r = x + 8$$

The height is:

$$h = 2x + 3$$

Expand the squared radius term

We square the radius expression:

$$r^2 = (x + 8)^2 = x^2 + 16x + 64$$

Multiply by the height expression

We multiply the squared radius by the height:

$$(x^2 + 16x + 64)(2x + 3)$$

Distributing each term:

$$= x^2(2x + 3) + 16x(2x + 3) + 64(2x + 3)$$

$$= 2x^3 + 3x^2 + 32x^2 + 48x + 128x + 192$$

Combine like terms

We group and sum the coefficients:

$$= 2x^3 + (3 + 32)x^2 + (48 + 128)x + 192$$

$$= 2x^3 + 35x^2 + 176x + 192$$

Multiply by pi for final volume

We multiply the entire polynomial by \(\pi\):

$$V = \pi(2x^3 + 35x^2 + 176x + 192)$$

$$V = 2\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi$$

This matches the third option.

Answer:

• (A) \(\pi x^3 + 19\pi x^2 + 112\pi x + 192\pi\)
• (B) 2\(\pi x^3 + 35\pi x^2 + 80\pi x + 48\pi\)
• (C) 2\(\pi x^3 + 35\pi x^2 + 176\pi x + 192\pi\) (Correct answer)
• (D) 4\(\pi x^3 + 44\pi x^2 + 105\pi x + 72\pi\)