To find the length of the ramp that will fit diagonally in the rectangular cage, we can use the 3-dimensional distance formula. The diagonal \( d \) of a rectangular prism can be calculated using the formula:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
where \( l \) is the length, \( w \) is the width, and \( h \) is the height of the cage.
Given:
- Length \( l = 70 \) (units)
- Width \( w = 35 \) (units)
- Height \( h = 50 \) (units)
Substituting these values into the formula:
\[ d = \sqrt{70^2 + 35^2 + 50^2} \]
Calculating the squares:
\[ 70^2 = 4900 \] \[ 35^2 = 1225 \] \[ 50^2 = 2500 \]
Now, summing these values:
\[ d = \sqrt{4900 + 1225 + 2500} = \sqrt{8625} \]
Now calculating \( \sqrt{8625} \):
\[ \sqrt{8625} \approx 92.9 \]
Thus, the length of the ramp should be approximately 92.9 units to fit diagonally in the cage.