Question

one rectangular solid with a square base has twice the height of another rectangular solid with a square base with the same side length. which statements about the two rectangular solids are true? check all that apply. the bases are congruent. the solids are similar. the ratio of the volumes of the first solid to the second solid is 8:1. the volume of the first solid is twice as much as the volume of the second solid. if the dimensions of the second solid are x by x by h, the first solid has 4xh more surface area than the second solid.

Explanation:

1. The bases are congruent: Both have square bases with the same side length, so their bases (squares) are congruent. Correct.
2. The solids are similar: For similar solids, all corresponding linear dimensions must be in proportion. Here, the base side lengths are equal (ratio 1:1) and heights are in ratio 2:1. Since not all linear ratios are equal, they are not similar. Incorrect.
3. Ratio of volumes 8:1: Volume of a rectangular solid is \( V = \text{base area} \times \text{height} \). Let the second solid have base side \( x \), height \( h \), so \( V_2 = x^2h \). First solid has height \( 2h \), so \( V_1 = x^2(2h) = 2x^2h \). Ratio \( V_1:V_2 = 2:1 \), not 8:1. Incorrect.
4. Volume of first is twice the second: As above, \( V_1 = 2x^2h \), \( V_2 = x^2h \), so \( V_1 = 2V_2 \). Correct.
5. Surface area difference: Surface area of a rectangular solid with square base (side \( s \), height \( H \)) is \( SA = 2s^2 + 4sH \). Second solid: \( SA_2 = 2x^2 + 4xh \). First solid (height \( 2h \)): \( SA_1 = 2x^2 + 4x(2h) = 2x^2 + 8xh \). Difference: \( SA_1 - SA_2 = (2x^2 + 8xh) - (2x^2 + 4xh) = 4xh \). Correct.

Answer:

• The bases are congruent.
• The volume of the first solid is twice as much as the volume of the second solid.
• If the dimensions of the second solid are \( x \) by \( x \) by \( h \), the first solid has \( 4xh \) more surface area than the second solid.