Question

which expression can be used to find the surface area of the prism?
$(3 \cdot 4) + (3 \cdot 3) + (3 \cdot 5) + 2\left(\frac{1}{2} \cdot 3 \cdot 4\
ight)$
what is the surface area of the prism?
s.a. = \boxed{38} in²

Explanation:

Step1: Recall Surface Area Formula for Triangular Prism

A triangular prism has two triangular bases and three rectangular faces. The formula for the surface area (SA) is \( SA = 2\times(\frac{1}{2}\times base\times height) + (length\times width_1) + (length\times width_2) + (length\times width_3) \), where the triangular base has base and height, and the rectangles have length (the length of the prism) and different widths (the sides of the triangle).

Step2: Identify Dimensions from the Prism

From the diagram, the triangular base has base \( 3 \) in and height \( 4 \) in. The length of the prism (the distance along the non - triangular faces) is \( 3 \) in. The sides of the triangular base: one side is \( 3 \) in, another is \( 4 \) in, and the hypotenuse (from the right triangle, using Pythagoras \( \sqrt{3^{2}+4^{2}} = 5 \)) is \( 5 \) in.

Step3: Substitute into the Surface Area Formula

• Area of two triangular bases: \( 2\times(\frac{1}{2}\times3\times4)= 3\times4 = 12 \) (we can also calculate it as \( 2\times(\frac{1}{2}\times3\times4)=12 \))
• Area of the three rectangular faces:
• First rectangle: \( 3\times4 \) (length \( 3 \), width \( 4 \))
• Second rectangle: \( 3\times3 \) (length \( 3 \), width \( 3 \))
• Third rectangle: \( 3\times5 \) (length \( 3 \), width \( 5 \))

Now, sum all these areas:
\( (3\times4)+(3\times3)+(3\times5)+2\times(\frac{1}{2}\times3\times4) \)
Calculate each term:

• \( 3\times4 = 12 \)
• \( 3\times3 = 9 \)
• \( 3\times5 = 15 \)
• \( 2\times(\frac{1}{2}\times3\times4)=12 \)

Sum them up: \( 12 + 9+15 + 12=48 \)? Wait, the given answer was 38, but let's re - check. Wait, maybe the length of the prism is 3, but maybe I misread the diagram. Wait, looking at the expression \( (3\cdot4)+(3\cdot3)+(3\cdot5)+2(\frac{1}{2}\cdot3\cdot4) \)
Let's calculate the expression:

• \( 3\times4=12 \)
• \( 3\times3 = 9 \)
• \( 3\times5=15 \)
• \( 2\times(\frac{1}{2}\times3\times4)=12 \)

Wait, \( 12 + 9+15 + 12=48 \). But the user - provided answer was 38, which is incorrect. Wait, maybe the length of the prism is 3, but the sides of the rectangles: maybe the length of the prism is 3, and the three rectangles have dimensions:
Wait, maybe the prism is a right triangular prism with the triangular base having legs 3 and 4, and the height (length of the prism) is 3.
The surface area formula for a triangular prism is \( SA=2B + Ph \), where \( B \) is the area of the base, \( P \) is the perimeter of the base, and \( h \) is the height of the prism.

• \( B=\frac{1}{2}\times3\times4 = 6 \)
• \( P=3 + 4+5=12 \)
• \( h = 3 \)

Then \( SA=2\times6+12\times3=12 + 36 = 48 \). But the given answer in the problem was 38, which is wrong. However, if we use the expression \( (3\cdot4)+(3\cdot3)+(3\cdot5)+2(\frac{1}{2}\cdot3\cdot4) \)
Calculate each part:
\( 3\times4 = 12 \), \( 3\times3=9 \), \( 3\times5 = 15 \), \( 2\times(\frac{1}{2}\times3\times4)=12 \)
\( 12+9 + 15+12=48 \)

Wait, maybe there was a misprint in the original problem's given answer of 38. But following the correct calculation with the given expression:

Answer:

The correct surface area of the prism is \( \boldsymbol{48} \) square inches. (If we follow the given wrong answer of 38, there is a mistake in the problem - setting. But based on the correct formula and the given expression, the surface area is 48.)