Question

question 1 (multiple choice worth 2 points)
(06.01 mc)
a rectangular pyramid has a height of 5 units and a volume of 50 units³. shannon states that a rectangular prism with the same base area and height has a volume that is three times the size of the given rectangular pyramid. which statement explains whether shannon is correct?
o a rectangular prism in which ba = 10 and h = 5 has a volume of 150 units³. therefore, shannon is correct
o a rectangular prism in which ba = 30 and h = 5 has a volume of 150 units³. therefore, shannon is correct
o a rectangular prism in which ba = 10 and h = 5 has a volume of 50 units³. therefore, shannon is incorrect
o a rectangular prism in which ba = 30 and h = 5 has a volume of 50 units³. therefore, shannon is incorrect

Explanation:

Step1: Recall Volume Formulas

The volume of a rectangular pyramid is \( V_{pyramid} = \frac{1}{3} \times BA \times h \), and the volume of a rectangular prism with the same base area (\( BA \)) and height (\( h \)) is \( V_{prism} = BA \times h \).

Step2: Calculate Pyramid Volume

Given the pyramid has \( h = 5 \) and \( V = 50 \). Using the pyramid volume formula:
\( 50 = \frac{1}{3} \times BA \times 5 \). Solve for \( BA \):
\( BA = \frac{50 \times 3}{5} = 30 \).

Step3: Calculate Prism Volume

For the prism with \( BA = 30 \) and \( h = 5 \), use \( V_{prism} = BA \times h \):
\( V_{prism} = 30 \times 5 = 150 \) units³.

Step4: Compare Volumes

The prism’s volume (\( 150 \)) is \( 150 \div 50 = 3 \) times the pyramid’s volume (\( 50 \)). So Shannon’s statement (prism volume is 3× pyramid) is correct when \( BA = 30 \), \( h = 5 \), and prism volume is \( 150 \).

Answer:

A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of \( 150 \) units³, therefore, Shannon is correct (the third option, e.g., if options are labeled as:
A. A rectangular prism in which \( BA = 10 \) and \( h = 5 \) has a volume of \( 150 \) units³, therefore, Shannon is correct
B. A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of \( 150 \) units³, therefore, Shannon is correct
C. A rectangular prism in which \( BA = 10 \) and \( h = 5 \) has a volume of \( 50 \) units³, therefore, Shannon is incorrect
D. A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of \( 50 \) units³, therefore, Shannon is incorrect

Then the answer is B. A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of \( 150 \) units³, therefore, Shannon is correct)