Right rectangular prism P has height x, length y, and width Right rectangular prism has height 2x. length 2y, and width 22. Which TWO of the following statements must be true?
The surface area of Q is 4 × the surface area of P, and the volume of Q is 8 × the volume of P.
To determine the surface area and volume of the right rectangular prisms P and Q, we use the formulas for each. Prism P has a height of x, length y, and width z; prism Q has dimensions that are double in height and length and a width of 22.
This statement is incorrect because when the dimensions of a prism are scaled, the surface area scales with the square of the scaling factor. Since the dimensions of Q are doubled in height and length, the surface area should actually be 4 times that of P, not 2.
Correct. The surface area of a rectangular prism is calculated as 2(lw + lh + wh). For Q, the dimensions are 2x, 2y, and 22, leading to a surface area that is 4 times that of P, confirming this statement.
This statement is false. The volume scales with the cube of the scaling factor. Since the height and length of Q are both doubled, the volume will be 2 × 2 × 1 (the width remains unchanged) = 4 times the volume of P, but this doesn't apply to O as it is not referenced correctly.
This statement is incorrect because the volume of a prism is calculated as length × width × height. Given the dimensions of Q, its volume is actually 8 times the volume of P, not 6.
Correct. The volume of prism Q, with dimensions doubled in height and length while width is constant, results in a volume that is 8 times greater than that of prism P, confirming this statement.
In evaluating the dimensions and resulting calculations for the right rectangular prisms P and Q, we find that the surface area of Q is 4 times that of P due to the scaling of dimensions, and the volume of Q is 8 times that of P, as it scales with the cube of the dimensions. Thus, statements B and E are validated as true based on geometric principles.
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