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Let $PQRSTUVW$ be a rectangular prism, as shown.  The area of face $TUVW$ is $10,$ the area of face $TUQP$ is $10,$ and the area of face $TPSW$ is $10.$ Find the volume of the rectangular prism.

 Dec 26, 2023
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First, we find the area of face $TSQP.$ Since the areas of faces $TUVW,$ $TUQP,$ and $TPSW$ are $10,$ $10,$ and $10,$ respectively, the length $TS$ is $\frac{10}{10} = 1.$

 

As a result, $TP = TW = 2,$ and the area of face $TSQP$ is $TP \cdot TQ = 2 \cdot 5 = 10$ (since $TQ = 5$).

 

Additionally, since the areas of faces $TUVW,$ $TUQP,$ and $TSQP$ are $10,$ $10,$ and $10,$ respectively, the width of the prism is $\frac{10}{1} = 10.$

Lastly, since the areas of faces $TUVW,$ $TSQP,$ and $TPSW$ are $10,$ $10,$ and $10,$ respectively, the height of the prism is $\frac{10}{2} = 5.$

As a result, the volume of the rectangular prism is $1 \cdot 10 \cdot 5 = \boxed{50}.$

 Dec 26, 2023

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