Question

7)9 km5 km7.8 km2709.654.58)10 yd10 yd11 yd10 yd10 yd9)2 yd5 yd3 yd4 yd10)11 m12 m12 m11 m12 m11)12 ft3 ft5 ft12 ft3 ft12)10 m10 m6.9 m

Explanation:

Problem 7: Right Hexagonal Prism

Step1: Find base area

A regular hexagon's area is $3 \times s \times a$, where $s=9$ km, $a=7.8$ km.
$\text{Base Area} = 3 \times 9 \times 7.8 = 210.6$ km²

Step2: Find lateral surface area

Lateral area = perimeter $\times$ height, perimeter $=6 \times 9=54$ km, height $=5$ km.
$\text{Lateral Area} = 54 \times 5 = 270$ km²

Step3: Total surface area

$\text{Total Area} = 2 \times \text{Base Area} + \text{Lateral Area}$
$\text{Total Area} = 2 \times 210.6 + 270 = 691.2$ km²

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Problem 8: Right Rectangular Prism

Step1: Identify dimensions

$l=10$ yd, $w=10$ yd, $h=11$ yd

Step2: Calculate total surface area

Formula: $2(lw + lh + wh)$
$\text{Total Area} = 2(10 \times 10 + 10 \times 11 + 10 \times 11)$
$\text{Total Area} = 2(100 + 110 + 110) = 2 \times 320 = 640$ yd²

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Problem 9: Right Trapezoidal Prism

Step1: Find trapezoid base area

Trapezoid area: $\frac{1}{2}(a + b)h$, $a=2$ yd, $b=4$ yd, height $=3$ yd.
$\text{Base Area} = \frac{1}{2}(2 + 4) \times 3 = 9$ yd²

Step2: Find lateral surface area

Sum of side lengths $\times$ prism length: $(2 + 3 + 4 + 5) \times \text{prism length}$ (prism length = height of trapezoid = 3 yd)
$\text{Lateral Area} = (2+3+4+5) \times 3 = 14 \times 3 = 42$ yd²

Step3: Total surface area

$\text{Total Area} = 2 \times 9 + 42 = 18 + 42 = 60$ yd²

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Problem 10: Right Rectangular Prism

Step1: Identify dimensions

$l=11$ m, $w=12$ m, $h=12$ m

Step2: Calculate total surface area

$\text{Total Area} = 2(lw + lh + wh)$
$\text{Total Area} = 2(11 \times 12 + 11 \times 12 + 12 \times 12)$
$\text{Total Area} = 2(132 + 132 + 144) = 2 \times 408 = 816$ m²

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Problem 11: Right Rectangular Prism

Step1: Identify dimensions

$l=12$ ft, $w=3$ ft, $h=5$ ft

Step2: Calculate total surface area

$\text{Total Area} = 2(lw + lh + wh)$
$\text{Total Area} = 2(12 \times 3 + 12 \times 5 + 3 \times 5)$
$\text{Total Area} = 2(36 + 60 + 15) = 2 \times 111 = 222$ ft²

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Problem 12: Right Pentagonal Prism

Step1: Find base area

Regular pentagon area: $\frac{5}{2} \times s \times a$, $s=10$ m, $a=6.9$ m.
$\text{Base Area} = \frac{5}{2} \times 10 \times 6.9 = 172.5$ m²

Step2: Find lateral surface area

Perimeter $\times$ height, perimeter $=5 \times 10=50$ m, height $=10$ m.
$\text{Lateral Area} = 50 \times 10 = 500$ m²

Step3: Total surface area

$\text{Total Area} = 2 \times 172.5 + 500 = 345 + 500 = 845$ m²

Answer:

1. $691.2$ km²
2. $640$ yd²
3. $60$ yd²
4. $816$ m²
5. $222$ ft²
6. $845$ m²