Question 35

The surface area of Q is 4 × the surface area of P and the volume of O is 8 × the volume of P.

The surface area of prism Q is indeed 4 times that of prism P due to its dimensions being doubled. Additionally, the volume of prism O is 8 times that of prism P, as it also involves a tripling of dimensions.

A) The surface area of Q is 2 × the surface area of P.

This statement is incorrect because the surface area of prism Q is calculated based on its dimensions, which are all doubled. Doubling each dimension results in a surface area that is quadrupled, not doubled.

B) The surface area of Q is 4 × the surface area of P.

This statement is correct. The surface area of a rectangular prism is calculated using the formula \(2(lw + lh + wh)\). If all dimensions are doubled, the resulting surface area increases by a factor of \(2^2 = 4\).

C) The volume of O is 4 × the volume of P.

This statement is incorrect. The volume of a rectangular prism is found by multiplying its dimensions. If all dimensions are doubled, the volume increases by a factor of \(2^3 = 8\), not 4.

D) The volume of Q is 6 × the volume of P.

This statement is incorrect. The volume of prism Q is calculated based on its dimensions, which are doubled. The correct factor for volume increase is 8, as it is proportional to the cube of the dimensions.

E) The volume of O is 8 × the volume of P.

This statement is correct. Since the dimensions of prism O are triple in one dimension and double in another, the volume is indeed \(2^3 = 8\) times that of prism P, which confirms this statement.

Conclusion

The correct statements regarding the surface area and volume are that the surface area of Q is 4 times that of P and the volume of O is 8 times that of P. All other options fail because they underestimate or miscalculate the effect of changing the dimensions of the prisms on their respective surface areas and volumes.