Question

find the surface area of each prism. round to the nearest tenth if necessary. show all work!
1.
2.
3.
4.
12 yd
10 yd
12 yd
6 m
12 m
8 m
6 in.
8 in.
10 in.
5 in.
7.8 cm
9 cm
9 cm
9 cm
12 cm
9 cm

Explanation:

1. Rectangular Prism (First Prism)

Step1: Recall the surface area formula for a rectangular prism

The formula for the surface area \(S\) of a rectangular prism is \(S = 2(lw+lh + wh)\), where \(l\) is the length, \(w\) is the width, and \(h\) is the height. Here, \(l = 12\) yd, \(w=10\) yd, \(h = 12\) yd.

Step2: Substitute the values into the formula

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2. Rectangular Prism (Second Prism)

Step1: Identify the dimensions

For the second rectangular - prism, \(l = 12\) m, \(w = 8\) m, \(h=6\) m.

Step2: Use the surface area formula \(S = 2(lw+lh + wh)\)

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3. Triangular Prism (Third Prism)

Step1: Find the area of the triangular bases

The base of the triangle: assume the triangle has sides \(a = 8\) in, \(b = 10\) in, \(c = 6\) in. Using Heron's formula \(s=\frac{a + b + c}{2}=\frac{8 + 10+6}{2}=12\) in. The area of the triangle \(A=\sqrt{s(s - a)(s - b)(s - c)}=\sqrt{12(12 - 8)(12 - 10)(12 - 6)}=\sqrt{12\times4\times2\times6}=\sqrt{576}=24\) in².

Step2: Find the area of the rectangular faces

The areas of the rectangular faces: \(A_1=10\times5 = 50\) in², \(A_2=8\times5=40\) in², \(A_3=6\times5 = 30\) in².

Step3: Calculate the surface area

The surface area \(S=2\times24+(50 + 40+30)=48 + 120=168\) in².

4. Triangular Prism (Fourth Prism)

Step1: Find the area of the triangular bases

The area of the triangular base \(A=\frac{1}{2}\times9\times7.8=35.1\) cm².

Step2: Find the area of the rectangular faces

The areas of the rectangular faces: \(A_1=9\times12 = 108\) cm², \(A_2=9\times12=108\) cm², \(A_3=9\times12 = 108\) cm².

Step3: Calculate the surface area

The surface area \(S=2\times35.1+(108+108 + 108)=70.2+324=394.2\) cm².

Answer:

1. \(768\) yd²
2. \(432\) m²
3. \(168\) in²
4. \(394.2\) cm²