Question

question 13 of 39
which two solid figures have the same volume?
a. a cone with a base of 6 cm² and a height of 12 cm
b. an oblique prism with a base of 6 cm² and a slant height of 12cm
c. an oblique solid with a base of 6 cm² and a height of 12 cm
d. a rectangular solid with a base of 6 cm² and a height of 12 cm

Explanation:

Step1: Recall volume formulas

The volume formula for a cone is \(V=\frac{1}{3}Bh\) (where \(B\) is the base area and \(h\) is the height). For a prism (including oblique prisms and rectangular solids) and other oblique solids with uniform - cross - section, the volume formula is \(V = Bh\) (where \(B\) is the base area and \(h\) is the perpendicular height).

Step2: Calculate volumes

• For option A: Given \(B = 6\mathrm{cm}^2\) and \(h=12\mathrm{cm}\), using the cone volume formula \(V_A=\frac{1}{3}\times6\times12 = 24\mathrm{cm}^3\)
• For option B: The formula for the volume of a prism is \(V = Bh\), where \(h\) is the perpendicular height. Since we are given the slant height (not the perpendicular height) for the oblique prism, we cannot calculate its volume using the formula \(V = Bh\) with the given information in the same way as for perpendicular - height - based volume calculations for prisms with known base area.
• For option C: Using the formula \(V = Bh\) (for a solid with uniform cross - section), with \(B = 6\mathrm{cm}^2\) and \(h = 12\mathrm{cm}\), we get \(V_C=6\times12=72\mathrm{cm}^3\)
• For option D: Using the formula \(V = Bh\) (for a rectangular solid, which is a type of prism), with \(B = 6\mathrm{cm}^2\) and \(h = 12\mathrm{cm}\), we get \(V_D=6\times12 = 72\mathrm{cm}^3\)

So, the oblique solid (option C) and the rectangular solid (option D) have the same volume.

Answer:

C. An oblique solid with a base of \(6\mathrm{cm}^2\) and a height of \(12\mathrm{cm}\), D. A rectangular solid with a base of \(6\mathrm{cm}^2\) and a height of \(12\mathrm{cm}\)