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Given:

The length of the entire rectangle is 10x + 7.

The width of the entire rectangle is 6x.

The length of the unshaded rectangle is 4x - 5.

The width of the unshaded rectangle is 2x.

We need to determine the area of the shaded region of the rectangle.

Area of the entire rectangle:

The area of the entire rectangle can be determined using the formula,

Substituting the values, we have;

Thus, the area of the entire rectangle is 60x² + 42x

Area of the unshaded rectangle:

The area of the unshaded rectangle can be determined using the formula,

Substituting the values, we have;

Thus, the area of the unshaded rectangle is 8x² - 10x

Area of the shaded region of the rectangle:

The area of the shaded region of the rectangle can be determined by subtracting the area of the entire rectangle by the area of the unshaded rectangle.

Thus, we have;

Thus, we have;

Thus, the area of the shaded region of the rectangle is 52x(x + 1)

576 cm^3

Step-by-step explanation:

Express this volume formula as V = π r^2 h, where the " ^ " symbol indicates exponentiation.  Substitute 4 for r (this is half the diameter), 12 cm for h, and 3 for π:  V = 3*(4)^2*12 cm^3, or V = 576 cm^3.