To find the length of the ramp that fits diagonally inside the rectangular prism cage, we can use the 3-dimensional Pythagorean theorem. The formula for the diagonal \(d\) of a rectangular prism is:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
where:
- \(l\) is the length,
- \(w\) is the width, and
- \(h\) is the height.
In this case, the dimensions of the cage are:
- \(l = 70\) cm,
- \(w = 35\) cm,
- \(h = 50\) cm.
Now, substituting the values into the formula:
\[ d = \sqrt{(70)^2 + (35)^2 + (50)^2} \]
Calculating each square:
\[ d = \sqrt{4900 + 1225 + 2500} \]
Now add these values together:
\[ 4900 + 1225 + 2500 = 8625 \]
Now take the square root:
\[ d = \sqrt{8625} \approx 92.9 \]
So, the length of the ramp that needs to fit diagonally in the cage is approximately 92.9 cm.