Question

the following rectangular pyramid has a height of 2 units and a volume of 18 cubic units. the oblique rectangular prism has a congruent base and the same height, and its slant height is 3 units.

what is the volume of the prism?

choose 1 answer:
a 6 cubic units
b 18 cubic units
c 36 cubic units
d 54 cubic units

Explanation:

Step1: Recall the volume formula for a pyramid

The volume of a rectangular pyramid is given by \( V_{pyramid} = \frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height.
We know \( V_{pyramid} = 18 \) cubic units and \( h = 2 \) units. So we can find \( B \) first.
From \( 18=\frac{1}{3}B\times2 \), we can solve for \( B \).
Multiply both sides by 3: \( 18\times3 = B\times2 \), so \( 54 = 2B \).
Then divide both sides by 2: \( B=\frac{54}{2}=27 \) square units.

Step2: Recall the volume formula for a prism

The volume of a rectangular prism (including oblique ones with the same base and height) is \( V_{prism}=Bh \).
We already found \( B = 27 \) square units and the height \( h = 2 \) units (same as the pyramid's height).
So \( V_{prism}=27\times2 = 54 \)? Wait, no, wait. Wait, the pyramid's volume is \( \frac{1}{3}Bh = 18 \), so \( Bh = 18\times3 = 54 \). And the prism's volume is \( Bh \), so the prism's volume is 54? Wait, but let's check again. Wait, the pyramid has volume \( \frac{1}{3}Bh = 18 \), so \( Bh = 54 \). The prism has the same base \( B \) and same height \( h \), so its volume is \( Bh = 54 \). Wait, but let's confirm the formula. The volume of a pyramid is one - third the volume of a prism with the same base and height. So if \( V_{pyramid}=\frac{1}{3}V_{prism} \), then \( V_{prism}=3V_{pyramid} \). Since \( V_{pyramid}=18 \), then \( V_{prism}=3\times18 = 54 \) cubic units.

Answer:

D. 54 cubic units