Question

calculate the surface area of each prism. show all y nearest tenth.

Explanation:

Step1: Identify the prism type

This is a triangular prism. The formula for the surface area of a triangular prism 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).

Step2: Calculate the area of the triangular base (\( B \))

The triangular base has a base of \( 12 \) in and a height of \( 8 \) in (from the right angle mark). The area of a triangle is \( \frac{1}{2} \times base \times height \).
\[
B=\frac{1}{2}\times12\times8 = 48 \text{ square inches}
\]

Step3: Calculate the perimeter of the triangular base (\( P \))

The triangular base has sides of \( 12 \) in, \( 10 \) in, and \( 10 \) in (the two equal sides, since it's an isosceles triangle? Wait, no, looking at the diagram, the sides are 12, 10, 10? Wait, the base is 12, and the other two sides are 10 each? Wait, the height is 8, so using Pythagoras, the legs: the base is 12, so half of it is 6, then the height is 8, so the hypotenuse (the equal sides) would be \( \sqrt{6^2 + 8^2}=\sqrt{36 + 64}=\sqrt{100}=10 \), yes, so the triangle is isosceles with sides 12, 10, 10. So perimeter \( P = 12 + 10 + 10 = 32 \) inches.

Step4: Identify the length of the prism (\( h \))

The length of the prism (the distance between the two triangular bases) is \( 14 \) in (from the diagram, the side of the rectangular face is 14 in).

Step5: Calculate the surface area using the formula \( SA = 2B + Ph \)

First, calculate \( 2B \): \( 2\times48 = 96 \)
Then, calculate \( Ph \): \( 32\times14 = 448 \)
Now, add them together: \( SA = 96 + 448 = 544 \) square inches. Wait, but wait, let's check again. Wait, maybe I made a mistake. Wait, the triangular base: base 12, height 8, so area is 48. Perimeter: 12 + 10 + 10 = 32. Length of prism is 14. So \( 2B = 96 \), \( Ph = 32*14 = 448 \), total \( 96 + 448 = 544 \). Wait, but let's confirm the formula. The surface area of a triangular prism is the sum of the areas of the two triangular bases and the three rectangular faces. The two triangular bases: 2(1/2128)=96. The three rectangular faces: one with dimensions 1214, and two with dimensions 1014. So area of rectangular faces: 1214 + 2(1014) = 168 + 280 = 448. Then total surface area: 96 + 448 = 544. Yes, that's correct.

Answer:

The surface area of the triangular prism is \(\boxed{544}\) square inches.