To find the length of the ramp that fits diagonally in the cage shaped as a rectangular prism, we can use the 3-dimensional diagonal formula for a rectangular prism:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
where:
- \(d\) is the diagonal length,
- \(l\) is the length,
- \(w\) is the width,
- \(h\) is the height.
In this case, the dimensions of the cage are:
- \(l = 70 \text{ cm}\)
- \(w = 35 \text{ cm}\)
- \(h = 50 \text{ cm}\)
Now, we can plug these values into the formula:
\[ d = \sqrt{70^2 + 35^2 + 50^2} \]
First, we calculate each term:
\[ 70^2 = 4900 \] \[ 35^2 = 1225 \] \[ 50^2 = 2500 \]
Now, we add these values together:
\[ d = \sqrt{4900 + 1225 + 2500} \] \[ d = \sqrt{8625} \]
Now we find the square root of 8625:
\[ d \approx 92.9 \]
Finally, rounding to the nearest tenth, the length of the ramp needed to fit diagonally in the cage is
\[ \boxed{92.9 \text{ cm}} \]