Question

question 1 (multiple choice worth 2 points)
(05.01 mc)
a rectangular prism 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 prism. 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:

1. First, recall the volume formula for a rectangular prism: \( V = BA \times h \), where \( BA \) is the base area and \( h \) is the height.
2. For the given prism: \( BA \) (let's find it first from the volume and height). Given \( V = 50 \) units³ and \( h = 5 \) units, so \( BA=\frac{V}{h}=\frac{50}{5} = 10 \) units².
3. Now, Shannon says a prism with the same base area (\( BA = 10 \)) and height (\( h = 5 \)) has three - times the volume? Wait, no, wait. Wait, no, the original prism has \( h = 5 \), \( V = 50 \), so \( BA = 10 \). Now, if we have a prism with \( BA=30 \) (three times the original \( BA = 10 \)) and \( h = 5 \), then its volume \( V'=30\times5 = 150 \), which is three times the original volume (\( 50\times3 = 150 \)). But wait, the options: Let's check each option:

• Option 1: \( BA = 10 \), \( h = 5 \), \( V=10\times5 = 50 \), not 150. So Shannon is wrong here.
• Option 2: \( BA = 30 \), \( h = 5 \), \( V = 30\times5=150 \), which is three times 50. So if Shannon's statement is about a prism with \( BA = 30 \) (three times original \( BA \)) and same height, then the volume is three times. Wait, the original prism: \( V = BA\times h=10\times5 = 50 \). A prism with \( BA = 30 \) (three times 10) and \( h = 5 \) has \( V = 30\times5 = 150 \), which is three times 50. So the correct option is the one where \( BA = 30 \), \( h = 5 \), volume 150, and Shannon is correct. Wait, let's re - read the options:
• Option 3: "A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of 150 units³, therefore, Shannon is correct" – this is correct. Because original \( BA=\frac{50}{5}=10 \), new \( BA = 30 \) (3 times 10), \( h = 5 \), \( V = 30\times5 = 150 \) (3 times 50).
• Wait, no, the original prism has \( h = 5 \), \( V = 50 \), so \( BA = 10 \). If we take a prism with \( BA = 30 \) (three times the original base area) and same height \( h = 5 \), then \( V=30\times5 = 150 \), which is three times the original volume (50). So the correct option is the third one? Wait, no, let's check the options again. Wait, the first option: "A rectangular prism in which \( BA = 10 \) and \( h = 5 \) has a volume of 150 units³, therefore, Shannon is correct" – no, \( 10\times5 = 50

eq150 \). The second option: "A rectangular prism in which \( BA = 10 \) and \( h = 5 \) has a volume of 50 units³, therefore, Shannon is incorrect" – but if Shannon's statement is about a prism with three - times the base area, this is not relevant. The third option: "A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of 150 units³, therefore, Shannon is correct" – this is correct. Because \( 30\times5 = 150 \), and 150 is three times 50 (the original volume). The fourth option: "A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of 50 units³, therefore, Shannon is incorrect" – no, \( 30\times5 = 150
eq50 \).
So the correct option is the one where \( BA = 30 \), \( h = 5 \), volume 150, and Shannon is correct. So the correct option is the third option (assuming the options are ordered as: 1. \( BA = 10 \), \( h = 5 \), \( V = 150 \), Shannon correct; 2. \( BA = 10 \), \( h = 5 \), \( V = 50 \), Shannon incorrect; 3. \( BA = 30 \), \( h = 5 \), \( V = 150 \), Shannon correct; 4. \( BA = 30 \), \( h = 5 \), \( V = 50 \), Shannon incorrect).

Answer:

The correct option is the one with the description: "A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of 150 units³, therefore, Shannon is correct" (assuming this is, for example, option C or the third option in the list of options provided in the question). If we assume the options are labeled as:

• Option 1: A rectangular prism in which \( BA = 10 \) and \( h = 5 \) has a volume of 150 units³, therefore, Shannon is correct
• Option 2: A rectangular prism in which \( BA = 10 \) and \( h = 5 \) has a volume of 50 units³, therefore, Shannon is incorrect
• Option 3: A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of 150 units³, therefore, Shannon is correct
• Option 4: A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of 50 units³, therefore, Shannon is incorrect

Then the answer is Option 3: A rectangular prism in which \( BA = 30 \) and \( h = 5 \) has a volume of 150 units³, therefore, Shannon is correct