Question

find the surface area and volume of the solid. round each measure to the nearest tenth, if necessary.
surface area:
cm²
volume:
cm³

Explanation:

Step1: Identify the solid (triangular prism)

The solid is a triangular prism with a triangular base (right triangle with legs 2.4 cm and 3.2 cm) and length (prism length) 4.0 cm, and the other side of the base triangle is 2.0 cm? Wait, no, let's re - examine. The triangular base: right triangle with base \(b = 3.2\) cm, height \(h=2.4\) cm. The length of the prism (the distance between the two triangular bases) is \(l = 4.0\) cm? Wait, no, the other edge is 2.0 cm? Wait, maybe the triangular base has sides: the right triangle has legs 2.4 and 3.2, hypotenuse? Wait, no, let's calculate the hypotenuse of the right triangle: \(\sqrt{2.4^{2}+3.2^{2}}=\sqrt{5.76 + 10.24}=\sqrt{16}=4\) cm? Wait, no, 2.4 squared is 5.76, 3.2 squared is 10.24, sum is 16, square root is 4. So the triangular base is a right triangle with legs 2.4 cm and 3.2 cm, hypotenuse 4 cm. And the length of the prism (the distance between the two triangular bases) is 2.0 cm? Wait, maybe I got the dimensions wrong. Let's re - define:

For a triangular prism, the surface area formula is \(SA=2B + Ph\), where \(B\) is the area of the triangular base, \(P\) is the perimeter of the triangular base, and \(h\) is the length of the prism (the distance between the two triangular bases). The volume formula is \(V = B\times h\), where \(B\) is the area of the triangular base and \(h\) is the length of the prism.

First, calculate the area of the triangular base \(B\):
\(B=\frac{1}{2}\times base\times height=\frac{1}{2}\times2.4\times3.2 = 3.84\) \(cm^{2}\)

Step2: Calculate the perimeter of the triangular base \(P\)

The sides of the triangular base: 2.4 cm, 3.2 cm, and 4.0 cm (since \(\sqrt{2.4^{2}+3.2^{2}} = 4\)). So \(P=2.4 + 3.2+4.0=9.6\) cm

Step3: Calculate the surface area

Assume the length of the prism (the distance between the two triangular bases) is \(l = 2.0\) cm. Then the lateral surface area is \(P\times l=9.6\times2.0 = 19.2\) \(cm^{2}\). The area of the two triangular bases is \(2\times B=2\times3.84 = 7.68\) \(cm^{2}\). So total surface area \(SA=19.2 + 7.68=26.88\approx26.9\) \(cm^{2}\) (wait, maybe the length of the prism is 4.0 cm? Let's re - check the diagram. If the triangular base has legs 2.4 and 3.2, and the other edge (the length of the prism) is 4.0 cm, and the other side of the base triangle is 2.0 cm? No, maybe I made a mistake. Let's try again.

Wait, maybe the triangular base is a right triangle with base 3.2 cm, height 2.4 cm, and the length of the prism (the distance along the direction perpendicular to the triangle) is 4.0 cm, and the other side of the base triangle is 2.0 cm? No, perhaps the correct dimensions are: triangular base: right triangle with legs \(a = 2.4\) cm, \(b = 3.2\) cm, length of prism \(l=2.0\) cm, and the hypotenuse of the triangle is 4 cm (as we calculated before).

Wait, let's recast:

Volume of triangular prism: \(V=\) area of base \(\times\) length of prism. Area of base \(B=\frac{1}{2}\times2.4\times3.2 = 3.84\) \(cm^{2}\). If the length of the prism is 4.0 cm, then \(V = 3.84\times4.0=15.36\approx15.4\) \(cm^{3}\). If the length of the prism is 2.0 cm, \(V = 3.84\times2.0 = 7.68\approx7.7\) \(cm^{3}\). Wait, the diagram shows 2.0 cm, 2.4 cm, 3.2 cm, 4.0 cm. Let's look at the diagram again: the triangular face has a right angle, with one leg 2.4 cm, another leg 3.2 cm, and the side of the prism (the length) is 4.0 cm, and the other edge of the base triangle is 2.0 cm? No, maybe the prism has a triangular base with sides 2.0 cm, 3.2 cm, and 2.4 cm? No, 2.0, 2.4, 3.2: check if it's a right triangle: \(2.0^{2}+2.4^{2}=4 + 5.76 = 9.76 eq3.2^{2}=10.24\). \(2.0^{2}+3.2^{2}=4+10.24 = 14.24 eq2.4^{2}=5.76\). \(2.4^{2}+3.2^{2}=5.76 + 10.24 = 16=4^{2}\). So the triangular base is a right triangle with legs 2.4 and 3.2, hypotenuse 4. And the length of the prism (the distance between the two triangular bases) is 2.0 cm.

So, perimeter of the triangular base \(P=2.4 + 3.2+4=9.6\) cm.

Area of triangular base \(B=\frac{1}{2}\times2.4\times3.2 = 3.84\) \(cm^{2}\)

Surface area: \(SA = 2B+P\times l\), where \(l = 2.0\) cm (length of prism)
\(SA=2\times3.84+9.6\times2.0=7.68 + 19.2 = 26.88\approx26.9\) \(cm^{2}\)

Volume: \(V=B\times l=3.84\times2.0 = 7.68\approx7.7\) \(cm^{3}\)

Wait, but maybe the length of the prism is 4.0 cm. Let's check the diagram again. The diagram has 2.4 cm, 3.2 cm, 2.0 cm, 4.0 cm. Maybe the triangular base is a right triangle with legs 2.4 and 2.0, and the length of the prism is 3.2? No, this is getting confusing. Wait, let's use the correct formula for a triangular prism:

If the triangular base is a right triangle with legs \(a\) and \(b\), and the height of the prism (the distance between the two bases) is \(c\), then:

- Area of base \(B=\frac{1}{2}ab\)
- Perimeter of base \(P=a + b+\sqrt{a^{2}+b^{2}}\)
- Surface area \(SA = 2B+P\times c\)
- Volume \(V=B\times c\)

From the diagram, let's assume \(a = 2.4\) cm, \(b = 3.2\) cm, \(c = 2.0\) cm (the 2.0 cm is the height of the prism). Then:

\(B=\frac{1}{2}\times2.4\times3.2 = 3.84\) \(cm^{2}\)

\(P=2.4 + 3.2+\sqrt{2.4^{2}+3.2^{2}}=2.4 + 3.2 + 4=9.6\) cm

\(SA=2\times3.84+9.6\times2.0=7.68 + 19.2 = 26.88\approx26.9\) \(cm^{2}\)

\(V=3.84\times2.0 = 7.68\approx7.7\) \(cm^{3}\)

Alternatively, if \(c = 4.0\) cm (the 4.0 cm is the height of the prism):

\(SA=2\times3.84+9.6\times4.0=7.68+38.4 = 46.08\approx46.1\) \(cm^{2}\)

\(V=3.84\times4.0 = 15.36\approx15.4\) \(cm^{3}\)

Wait, the diagram shows 2.0 cm, 2.4 cm, 3.2 cm, 4.0 cm. Let's look at the labels: the triangular face has a right angle, with one leg 2.4 cm, another leg 3.2 cm, and the side of the prism (the length) is 4.0 cm, and the other edge of the base triangle is 2.0 cm? No, maybe the prism is a triangular prism with a triangular base that has sides 2.0 cm, 3.2 cm, and 2.4 cm, but that's not a right triangle. Wait, the right angle is marked, so it's a right triangle. So the legs are 2.4 and 3.2, hypotenuse 4, and the length of the prism is 2.0 cm. So:

Surface area: \(2\times(\frac{1}{2}\times2.4\times3.2)+(2.4 + 3.2+4)\times2.0=7.68 + 19.2 = 26.88\approx26.9\) \(cm^{2}\)

Volume: \(\frac{1}{2}\times2.4\times3.2\times2.0=7.68\approx7.7\) \(cm^{3}\)

Answer:

Surface area: \(\approx26.9\) \(cm^{2}\)
Volume: \(\approx7.7\) \(cm^{3}\)