Question

find the surface area and volume of the solid. round each measure to the nearest tenth, if necessary. need help with this question? volume. cm³ surface area cm²

Explanation:

Step1: Find the base - area of the triangular base

The base is a right - triangle with legs \(a = 2.0\mathrm{cm}\) and \(b = 2.4\mathrm{cm}\). The area of a right - triangle \(A_{base}=\frac{1}{2}\times a\times b\).
\[A_{base}=\frac{1}{2}\times2.0\times2.4 = 2.4\mathrm{cm}^2\]

Step2: Calculate the volume of the prism

The volume of a prism \(V=A_{base}\times h\), where \(h = 3.2\mathrm{cm}\).
\[V = 2.4\times3.2=7.68\mathrm{cm}^3\approx7.7\mathrm{cm}^3\]

Step3: Find the lengths of the sides of the triangular base

Using the Pythagorean theorem, the hypotenuse \(c\) of the right - triangle base: \(c=\sqrt{2.0^{2}+2.4^{2}}=\sqrt{4 + 5.76}=\sqrt{9.76}\approx3.12\mathrm{cm}\)

Step4: Calculate the surface area of the prism

The surface area of a triangular prism \(S = 2A_{base}+(a + b + c)h\).
\[S=2\times2.4+(2.0 + 2.4+3.12)\times3.2\]
\[S = 4.8+(7.52)\times3.2\]
\[S = 4.8+24.064\]
\[S=28.864\mathrm{cm}^2\approx28.9\mathrm{cm}^2\]

Answer:

Volume: \(7.7\mathrm{cm}^3\)
Surface Area: \(28.9\mathrm{cm}^2\)