Question

question 5 (multiple choice worth 2 points) (03.03 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? a rectangular prism in which ( ba = 10 ) and ( h = 5 ) has a volume of 150 units³. therefore, shannon is correct. a rectangular prism in which ( ba = 30 ) and ( h = 5 ) has a volume of 150 units³. therefore, shannon is correct. a rectangular prism in which ( ba = 10 ) and ( h = 5 ) has a volume of 50 units³. therefore, shannon is incorrect. 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 formula for a rectangular pyramid is $V_{pyramid} = \frac{1}{3} \times BA \times h$, where $BA$ is the base area and $h$ is the height. The volume formula for a rectangular prism is $V_{prism} = BA \times h$.

Step2: Find Base Area of Pyramid

Given the pyramid has $h = 5$ units and $V_{pyramid}=50$ units³. Using $V_{pyramid}=\frac{1}{3} \times BA \times h$, we substitute values:
$50=\frac{1}{3} \times BA \times 5$. Solving for $BA$:
Multiply both sides by 3: $150 = BA \times 5$
Divide by 5: $BA = 30$? Wait, no—wait, wait, let's recalculate. Wait, $50=\frac{1}{3} \times BA \times 5$ → $BA=\frac{50\times3}{5}=30$? Wait, no, wait the options: Wait, maybe I made a mistake. Wait, the pyramid's volume is 50, height 5. So $V_{pyramid}=\frac{1}{3}BAh$ → $50=\frac{1}{3}BA(5)$ → $BA=\frac{50\times3}{5}=30$. Wait, but the options have BA=10 or 30. Wait, maybe the first step: Let's check the prism's volume. The prism with same base area and height as the pyramid (pyramid's BA: from $V_{pyramid}=50=\frac{1}{3}BA\times5$ → $BA = \frac{50\times3}{5}=30$? Wait, no, wait, no—wait, the pyramid's volume is 50, so if we take the prism with same BA and h, then $V_{prism}=BA\times h$. But the pyramid's BA: from $V_{pyramid}=\frac{1}{3}BAh$ → $BA=\frac{3V_{pyramid}}{h}=\frac{3\times50}{5}=30$. So the prism with BA=30 and h=5 would have $V_{prism}=30\times5=150$, which is 3×50=150. But wait, the first option says BA=10. Wait, maybe I messed up. Wait, let's re-express: The pyramid's volume is 50, so 3×50=150. So the prism's volume should be 150. Now, check the options:

Option 1: Prism with BA=10, h=5: volume=10×5=50≠150.

Option 2: Prism with BA=30, h=5: volume=30×5=150=3×50. So that's correct. Wait, but wait, the pyramid's BA: from $V_{pyramid}=\frac{1}{3}BAh$ → $BA=\frac{3V}{h}=\frac{3×50}{5}=30$. So the prism with same BA (30) and h (5) has volume 30×5=150, which is 3×50. So the correct option is the one where BA=30, h=5, volume 150, so Shannon is correct. Wait, but let's check the options again:

Option A: A rectangular prism in which BA = 10 and h = 5 has a volume of 150 units³. Therefore, Shannon is correct. → BA=10, h=5: volume=10×5=50≠150. So A is wrong.

Option B: A rectangular prism in which BA = 30 and h = 5 has a volume of 150 units³. Therefore, Shannon is correct. → 30×5=150, which is 3×50 (pyramid's volume). So this works.

Option C: A rectangular prism in which BA = 10 and h = 5 has a volume of 50 units³. Therefore, Shannon is incorrect. → 10×5=50, but the pyramid's BA is 30 (from $V_{pyramid}=50=\frac{1}{3}BA×5$ → BA=30), so the prism should have BA=30, not 10. So C is wrong.

Option D: A rectangular prism in which BA = 30 and h = 5 has a volume of 50 units³. Therefore, Shannon is incorrect. → 30×5=150≠50, so D is wrong.

Wait, but the first step: The pyramid's BA is 30 (from $V_{pyramid}=\frac{1}{3}BAh$ → $BA=\frac{3×50}{5}=30$). So the prism with same BA (30) and h (5) has volume 30×5=150, which is 3×50. So Shannon says the prism's volume is three times the pyramid's, which is 50×3=150. So the correct statement is the one where BA=30, h=5, volume 150, so Shannon is correct. So the correct option is B.

Answer:

B. A rectangular prism in which \( \text{BA} = 30 \) and \( h = 5 \) has a volume of \( 150 \) units\(^3\). Therefore, Shannon is correct.