Question

a right rectangular prism has base dimensions of 3 inches by 12 inches. an oblique rectangular prism has base dimensions of 4 inches by 9 inches. if the prisms are the same height, how do their volumes compare? the volumes are equal, because the bases are congruent. the volumes are equal, because the heights are equal and the horizontal cross - sectional areas at every level are also equal. the volumes are not equal, because their horizontal cross - sectional areas are not the same at every level.

Explanation:

Step1: Calculate base area of right prism

$\text{Area}_1 = 12 \times 3 = 36$ square inches

Step2: Calculate base area of oblique prism

$\text{Area}_2 = 9 \times 4 = 36$ square inches

Step3: Recall prism volume formula

Volume of a prism: $V = \text{Base Area} \times \text{Height}$

Step4: Compare volumes with equal height

Let height = $h$.
Right prism volume: $V_1 = 36h$
Oblique prism volume: $V_2 = 36h$
By Cavalieri's principle, cross-sectional areas at every horizontal level are equal for equal height, so volumes are equal.

Answer:

The volumes are equal, because the heights are equal and the horizontal cross-sectional areas at every level are also equal.