Question

Melissa has a gerbil that lives in a cage in the shape of a rectangular prism. She wants to put a ramp in the cage to give her gerbil exercise. The cage has a length of 70 cm, a width of 35 cm, and a height of 50 cm. How long does the ramp need to be to fit diagonally in the cage? Round the answer to the nearest tenth. (1 point)
• 78.6 cm
• 92.9 cm
• 60.2 cm
• 116.3 cm

Answer 1

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:

\[
\boxed{92.9 \, \text{cm}}
\]