To find the length of the ramp that fits diagonally in the cage, we can use the formula for the diagonal of a rectangular prism:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
Where:
- \( l \) = length of the cage
- \( w \) = width of the cage
- \( h \) = height of the cage
Substituting the values:
- \( l = 70 \) cm
- \( w = 35 \) cm
- \( h = 50 \) cm
Now we calculate the diagonal \( d \):
\[ d = \sqrt{(70)^2 + (35)^2 + (50)^2} \]
Calculating each square:
\[ (70)^2 = 4900 \] \[ (35)^2 = 1225 \] \[ (50)^2 = 2500 \]
Now sum these values:
\[ 4900 + 1225 + 2500 = 8625 \]
Now take the square root:
\[ d = \sqrt{8625} \approx 92.9 \text{ cm} \]
Thus, the length of the ramp needs to be approximately 92.9 cm.