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# geometry

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Let PQRSTUVW be a rectangular prism, as shown. The area of face PQRS is 6. The area of face PQUT is 15. The area of face QUVR is 10. Find the volume of the rectangular prism.

Jun 12, 2021

#1
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This problem can be solved through guessing.

We know that one side length of each side corresponds to one side length of another side.

Thus, we can split $6 = 3 \cdot 2$. From here, we know that the $2$ must correspond to the side with an area of $10$, as $2$ can't divide $15$.

As a result, the measurements of the side with an area of $10$ are $5 \cdot 2$.

Finally, the side lengths of the side with an area of $15$ are $5 \cdot 3$, which also corresponds to what we found previously.

Our 3 measurements are $2, 3, 5$, meaning the volume is $2 \cdot 3 \cdot 5 = 30$.

Jun 12, 2021
#2
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Let PQRSTUVW be a rectangular prism, as shown. The area of face PQRS is 6. The area of face PQUT is 15. The area of face QUVR is 10. Find the volume of the rectangular prism.

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[PQRS] = 6           [PQUT] = 15            [QUVR] = 10

(xz) = 6         x = 6/z          z = 6/x

(yz) = 15       y = 15/z        z = 15/y

(xy) = 10       x = 10/y        y = 10/x

(yz) = (10/x)(6/x) = 15        x = 2          y = 10/2         z = 6/2

V = xyz

Jun 12, 2021