1. **Problem 1: Cube with edge length 4 1/2 in** - Given: edge length $s = 4 \frac{1}{2} = 4.5$ in - Formula for surface area of a cube: $$SA = 6s^2$$ - Calculate area of one face: $$A = s^2 = 4.5^2 = 20.25$$ - Calculate total surface area: $$SA = 6 \times 20.25 = 121.5 \text{ in}^2$$ 2. **Problem 2: Triangular prism with sides 18 ft, 15 ft, 10 ft and height 11.2 ft** - Given: triangle sides $a=18$, $b=15$, $c=10$ ft, height $h=11.2$ ft - Find area of triangular base using Heron's formula: - Semi-perimeter: $$s = \frac{18 + 15 + 10}{2} = 21.5$$ - Area: $$A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{21.5(21.5-18)(21.5-15)(21.5-10)}$$ - Calculate inside: $$= \sqrt{21.5 \times 3.5 \times 6.5 \times 11.5} = \sqrt{5623.56} \approx 75.0$$ - Surface area formula for prism: $$SA = 2A + Ph$$ where $P$ is perimeter - Perimeter: $$P = 18 + 15 + 10 = 43$$ - Calculate surface area: $$SA = 2 \times 75 + 43 \times 11.2 = 150 + 481.6 = 631.6 \text{ ft}^2$$ 3. **Problem 3: Rectangular prism with dimensions 7 cm, 11 cm, 3 cm** - Given: length $l=7$, width $w=11$, height $h=3$ cm - Surface area formula: $$SA = 2(lw + lh + wh)$$ - Calculate each area: - $lw = 7 \times 11 = 77$ - $lh = 7 \times 3 = 21$ - $wh = 11 \times 3 = 33$ - Sum: $$77 + 21 + 33 = 131$$ - Calculate surface area: $$SA = 2 \times 131 = 262 \text{ cm}^2$$ 4. **Problem 4: Rectangular prism with dimensions 8 m, 10 m, 23 m** - Given: $l=8$, $w=10$, $h=23$ m - Surface area formula: $$SA = 2(lw + lh + wh)$$ - Calculate each area: - $lw = 8 \times 10 = 80$ - $lh = 8 \times 23 = 184$ - $wh = 10 \times 23 = 230$ - Sum: $$80 + 184 + 230 = 494$$ - Calculate surface area: $$SA = 2 \times 494 = 988 \text{ m}^2$$ 5. **Problem 5: Triangular prism with sides 19 mm, 21 mm, 13.1 mm and height 15 mm, 9 mm** - Given triangle sides $a=19$, $b=21$, $c=13.1$ mm, prism height $h=15$ mm, and another dimension 9 mm (assumed prism length) - Calculate area of triangular base using Heron's formula: - Semi-perimeter: $$s = \frac{19 + 21 + 13.1}{2} = 26.05$$ - Area: $$A = \sqrt{26.05(26.05-19)(26.05-21)(26.05-13.1)}$$ - Calculate inside: $$= \sqrt{26.05 \times 7.05 \times 5.05 \times 12.95} = \sqrt{12007.5} \approx 109.6$$ - Perimeter: $$P = 19 + 21 + 13.1 = 53.1$$ - Surface area formula: $$SA = 2A + Ph$$ where $h=15$ mm - Calculate surface area: $$SA = 2 \times 109.6 + 53.1 \times 15 = 219.2 + 796.5 = 1015.7 \text{ mm}^2$$