To determine which rectangular prism has the longer diagonal, we'll first calculate the diagonal of the base for each prism using the Pythagorean Theorem, and then use that result to calculate the overall diagonal of the prism.

Step 1: Use the Pythagorean Theorem for the Base Diagonal (c)

The base of a rectangular prism is a rectangle, and the diagonal $c$ of that rectangle can be calculated using the Pythagorean Theorem:

$c = \sqrt{a^2 + b^2}$

Where:

• $a$ and $b$ are the lengths of the sides of the rectangular base.

Step 2: Calculate the Overall Diagonal of the Prism (d)

Once we know the base diagonal $c$, the overall diagonal $d$ of the rectangular prism (which includes the height) can be found using the 3D Pythagorean Theorem:

$d = \sqrt{c^2 + h^2}$

Where:

• $c$ is the base diagonal.
• $h$ is the height of the prism.

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Prism P:

• Base dimensions: $15$ (length), $4$ (width)
• Height: $9$

Step 1: Calculate the base diagonal $c$ of Prism P:

$c = \sqrt{15^2 + 4^2} = \sqrt{225 + 16} = \sqrt{241} \approx 15.5$

Step 2: Calculate the overall diagonal $d$ of Prism P:

$d = \sqrt{15.5^2 + 9^2} = \sqrt{240.25 + 81} = \sqrt{321.25} \approx 17.9$

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Prism Q:

• Base dimensions: $12$ (length), $4$ (width)
• Height: $4$

Step 1: Calculate the base diagonal $c$ of Prism Q:

$c = \sqrt{12^2 + 4^2} = \sqrt{144 + 16} = \sqrt{160} \approx 12.6$

Step 2: Calculate the overall diagonal $d$ of Prism Q:

$d = \sqrt{12.6^2 + 4^2} = \sqrt{158.76 + 16} = \sqrt{174.76} \approx 13.2$

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Conclusion:

• Prism P has a diagonal of approximately 17.9 units.
• Prism Q has a diagonal of approximately 13.2 units.

Thus, Prism P has the longer diagonal.

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Would you like a further explanation or assistance with another problem?

Additional Questions:

1. How can the Pythagorean Theorem be applied to 3D shapes?
2. What happens to the diagonal if the height of the prism is increased or decreased?
3. How do you calculate the surface area of a rectangular prism?
4. What are the differences between a cube and a rectangular prism?
5. How do you find the volume of a rectangular prism?

Tip:

Always double-check the values you use for the dimensions when applying the Pythagorean Theorem—small mistakes in measurements can lead to large differences in the final answer.