What is the volume in cubic inches of a right, rectangular prism with side, front and bottom faces having an area 15 square inches, 10 square inches and 6 square inches, respectively?
Let the sides of the prism have lengths $x$, $y$, and $z$. We have the equations $xy=15$, $yz=10$ and $xz=6$. Multiplying these equations together, we have $xy\cdot yz \cdot xz = 15\cdot10\cdot6 \Rightarrow x^2y^2z^2=900$. Since the volume of the prism is equal to $xyz$, we take the square root of both sides to get the volume as $\sqrt{900}=\boxed{30}$ cubic inches.
30
Level 2
Geometry
Problem: What is the volume in cubic inches of a right, rectangular prism with side, front and bottom faces having an area 15 square inches, 10 square inches and 6 square inches, respectively?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the volume of the right rectangular prism, we need to determine the dimensions of the prism based on the areas of its faces. Let the dimensions of the prism be denoted as length (l), width (w), and height (h).

The areas of the three faces provide us with the following equations:
1. Area of the side face (l * h) = 15 square inches
2. Area of the front face (w * h) = 10 square inches
3. Area of the bottom face (l * w) = 6 square inches

From these equations, we can express the dimensions in terms of one variable. 

First, we can express h from the first equation:
h = 15 / l

Next, we can express h from the second equation:
h = 10 / w

Setting these two expressions for h equal gives us:
15 / l = 10 / w

Cross-multiplying leads to:
15w = 10l
This simplifies to:
l = (3/2)w

Now we can substitute l in terms of w into the third equation:
l * w = 6
(3/2)w * w = 6
(3/2)w^2 = 6

Multiplying both sides by 2 to eliminate the fraction:
3w^2 = 12
Dividing by 3:
w^2 = 4
Taking the square root:
w = 2

Now substituting w back to find l:
l = (3/2) * 2 = 3

Next, we substitute w back to find h using the second equation:
h = 10 / w = 10 / 2 = 5

Now we have the dimensions of the prism:
l = 3, w = 2, h = 5

Finally, we calculate the volume V of the prism using the formula:
V = l * w * h
V = 3 * 2 * 5 = 30 cubic inches

Thus, the final answer is 30.<|END_OF_LATENT|> The final answer is $30$.