To find the length of the ramp that fits diagonally in the cage, we need to calculate the space diagonal of the rectangular prism (cage). The formula for the space diagonal \(d\) of a rectangular prism with length \(l\), width \(w\), and height \(h\) is:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
Given:
- Length \(l = 70 , \text{cm}\)
- Width \(w = 35 , \text{cm}\)
- Height \(h = 50 , \text{cm}\)
Now we plug in the values:
\[ d = \sqrt{(70)^2 + (35)^2 + (50)^2} \]
Calculating each term:
\[ 70^2 = 4900 \] \[ 35^2 = 1225 \] \[ 50^2 = 2500 \]
Now sum them up:
\[ d = \sqrt{4900 + 1225 + 2500} = \sqrt{8625} \]
Now we calculate \(\sqrt{8625}\):
\[ d \approx 92.9 , \text{cm} \]
Rounding to the nearest tenth, the length of the ramp needed is:
\[ \boxed{92.9 , \text{cm}} \]
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