Question

a hollow post is made out of a lightweight aluminum alloy that is 1.5 cm thick. the alloy is folded into the shape of a right rectangular prism measuring 12 cm by 10 cm by 30 cm. find the total volume of alloy in the post. round your answer to the nearest hundredth if necessary. (note: diagram is not drawn to scale.)

Explanation:

Step1: Calculate the outer volume

The formula for the volume of a rectangular prism is \(V = l\times w\times h\).
For the outer prism, \(l = 12\) cm, \(w = 10\) cm, \(h = 30\) cm.
So, \(V_{outer}=12\times10\times30 = 3600\) \(cm^{3}\).

Step2: Calculate the inner dimensions

Since the alloy is \(1.5\) cm thick, the inner length \(l_{inner}=12 - 2\times1.5=9\) cm, the inner width \(w_{inner}=10 - 2\times1.5 = 7\) cm, and the height remains the same \(h_{inner}=30\) cm.

Step3: Calculate the inner volume

Using the volume formula \(V = l\times w\times h\) again, \(V_{inner}=9\times7\times30=1890\) \(cm^{3}\).

Step4: Calculate the volume of the alloy

The volume of the alloy \(V = V_{outer}-V_{inner}\).
\(V = 3600 - 1890=1710\) \(cm^{3}\).

Answer:

\(1710\) \(cm^{3}\)