To find the length of the ramp that fits diagonally in the cage (which is a rectangular prism), we can use the 3D distance formula. The formula for the diagonal \(d\) of a rectangular prism is given by:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
where \(l\) is the length, \(w\) is the width, and \(h\) is the height of the prism.
Given:
- Length \(l = 70\) cm
- Width \(w = 35\) cm
- Height \(h = 50\) cm
Now, we will substitute the values into the formula:
\[ d = \sqrt{(70)^2 + (35)^2 + (50)^2} \]
Calculating each squared term:
\[ 70^2 = 4900 \] \[ 35^2 = 1225 \] \[ 50^2 = 2500 \]
Now, add these values together:
\[ 4900 + 1225 + 2500 = 8625 \]
Now, find the square root of 8625:
\[ d = \sqrt{8625} \approx 92.9 \text{ cm} \]
Rounding to the nearest tenth results in:
\[ d \approx 92.9 \text{ cm} \]
Therefore, the length of the ramp that needs to fit diagonally in the cage is 92.9 cm.