The fourth expression, 2πx³ - 12πx² - 24πx + 63π, accurately represents the volume of the cylinder in the image .
The fourth expression is indeed the correct one that represents the volume of the cylinder. Here's why:
Formula: The volume of a cylinder is calculated by the formula V = πr²h, where V is the volume, π (pi) is a mathematical constant roughly equal to 3.14, r is the radius of the cylinder's base, and h is the height of the cylinder.
Information in the image: The image provides the following information:
r = x - 3 (radius is expressed in terms of x)
h = 2x + 7 (height is expressed in terms of x)
Substituting values: Substituting the expressions for r and h in the formula V = πr²h, we get: V = π(x - 3)² (2x + 7) Expanding the parentheses and simplifying the expression, we obtain: V = 2πx³ - 12πx² - 24πx + 63π
Therefore, the fourth expression, 2πx³ - 12πx² - 24πx + 63π, accurately represents the volume of the cylinder in the image.
It's important to note that the other expressions provided in the image are incorrect:
The second expression, 2πx³ - 5πx² - 24πx + 63π, has a different coefficient for the x² term.
The third expression, 2πx³ + 7πx² - 18πx - 63π, has incorrect coefficients for both the x² and x terms.
The first expression, (2πx³ -12 πx² - 24πx+63π), has incorrect coefficients for the x and x² terms.