1. **Problem 11:** A cone with diameter 10 m has surface area 290.6 m². Find its slant height. 2. **Formula for surface area of a cone:** $$SA = \pi r^2 + \pi r l$$ where $r$ is radius, $l$ is slant height. 3. Given diameter = 10 m, so radius $r = \frac{10}{2} = 5$ m. 4. Substitute known values: $$290.6 = \pi (5)^2 + \pi (5) l$$ $$290.6 = 25\pi + 5\pi l$$ 5. Isolate $l$: $$290.6 - 25\pi = 5\pi l$$ 6. Calculate numeric values: $$25\pi \approx 78.54$$ $$290.6 - 78.54 = 212.06$$ 7. Divide both sides by $5\pi$: $$l = \frac{212.06}{5\pi}$$ $$l = \frac{212.06}{15.707}$$ 8. Simplify with cancellation: $$l = \frac{\cancel{212.06}}{\cancel{15.707}} \approx 13.5$$ **Answer:** The slant height $l \approx 13.5$ meters. --- 1. **Problem 12:** Regular hexagonal pyramid with base area 166.28 ft², slant height 9.8 ft, surface area 401.48 ft². Find side length of base. 2. **Formula for surface area of pyramid:** $$SA = B + \frac{1}{2} P l$$ where $B$ is base area, $P$ is perimeter, $l$ is slant height. 3. Given: $$B = 166.28, \quad l = 9.8, \quad SA = 401.48$$ 4. Substitute values: $$401.48 = 166.28 + \frac{1}{2} P (9.8)$$ 5. Isolate perimeter $P$: $$401.48 - 166.28 = 4.9 P$$ $$235.2 = 4.9 P$$ 6. Divide both sides by 4.9: $$P = \frac{235.2}{4.9}$$ 7. Simplify with cancellation: $$P = \frac{\cancel{235.2}}{\cancel{4.9}} \approx 48$$ 8. For a regular hexagon, perimeter $P = 6s$, where $s$ is side length. 9. Solve for $s$: $$s = \frac{P}{6} = \frac{48}{6} = 8$$ **Answer:** Side length of base $s = 8$ feet. --- 1. **Problem 13:** Triangular pyramid with equilateral base side length 10 cm, surface area 214.5 cm². Find slant height. 2. **Base area of equilateral triangle:** $$B = \frac{\sqrt{3}}{4} s^2$$ 3. Calculate base area: $$B = \frac{\sqrt{3}}{4} (10)^2 = \frac{\sqrt{3}}{4} \times 100 = 25\sqrt{3} \approx 43.3$$ 4. Surface area formula for pyramid: $$SA = B + \frac{1}{2} P l$$ where $P$ is perimeter, $l$ is slant height. 5. Perimeter of equilateral triangle: $$P = 3 \times 10 = 30$$ 6. Substitute known values: $$214.5 = 43.3 + \frac{1}{2} (30) l$$ 7. Simplify: $$214.5 - 43.3 = 15 l$$ $$171.2 = 15 l$$ 8. Divide both sides by 15: $$l = \frac{171.2}{15}$$ 9. Simplify with cancellation: $$l = \frac{\cancel{171.2}}{\cancel{15}} \approx 11.41$$ **Answer:** Slant height $l \approx 11.41$ cm.