Question

the drawing below shows two similar right triangular prisms. the triangular bases of the smaller prism each have a hypotenuse with a length of 5 dekameters and sides with lengths of 3 dekameters and 4 dekameters. the ratio of the smaller prisms hypotenuse to the larger prisms hypotenuse is \\( \frac { 5 } { 15 } \\).
what is the volume, \\( v \\), of the larger right triangular prism?
5 of 9 questions
\\( v = 450 \mathrm { cu } \mathrm { dm } \\)
\\( v = 1620 \mathrm { cu } \mathrm { dm } \\)
\\( v = 60 \mathrm { cu } \mathrm { dm } \\)
\\( v = 120 \mathrm { cu } \mathrm { dm } \\)

Explanation:

Step1: Calculate the volume of the smaller prism

The formula for the volume of a triangular prism is \(V = \text{Base Area}\times\text{Height}\).
The base of the smaller prism is a right - triangle with legs \(a = 3\) dm and \(b = 4\) dm. The area of the base \(B=\frac{1}{2}\times3\times4=6\) square dm. The height (length) of the smaller prism \(h = 10\) dm.
So, the volume of the smaller prism \(V_{s}=6\times10 = 60\) cu dm.

Step2: Use the ratio of similarity

The ratio of the hypotenuses of the two similar prisms is \(\frac{5}{15}=\frac{1}{3}\). For similar solids, if the ratio of corresponding linear dimensions (such as the hypotenuse of the base triangle) is \(k=\frac{1}{3}\), the ratio of their volumes is \(k^{3}\). Let the volume of the larger prism be \(V_{l}\). Then \(\frac{V_{s}}{V_{l}}=(\frac{1}{3})^{3}=\frac{1}{27}\).

Step3: Solve for the volume of the larger prism

We know \(V_{s} = 60\) cu dm and \(\frac{V_{s}}{V_{l}}=\frac{1}{27}\). Cross - multiplying gives \(V_{l}=V_{s}\times27\).
Another way:
The base of the larger prism:
Since the ratio of the hypotenuses of the base triangles (similarity ratio) is \(\frac{5}{15}=\frac{1}{3}\), the legs of the base triangle of the larger prism are \(3\times3 = 9\) dm and \(4\times3=12\) dm. The area of the base of the larger prism \(B_{l}=\frac{1}{2}\times9\times12 = 54\) square dm. The height (length) of the larger prism \(h_{l}=30\) dm.
Using the formula \(V=\text{Base Area}\times\text{Height}\), \(V = 54\times30=1620\) cu dm.

Answer:

\(V = 1620\) cu dm