The band, Limits to Infinity, requests two cylindrical towers as stage decorations. Both cylindrical towers have the same total surface area but different heights. Total surface area = 120 m²

Let $r_A$ be the radius of tower A and $h_A$ be the height of tower A. Let $r_B$ be the radius of tower B and $h_B$ be the height of tower B. Given: 1) Area of base of tower A: $\pi r_A^2 = 16.6$ m$^2$. 2) Total surface area for both: $S = 2\pi r^2 + 2\pi r h = 120$ m$^2$. 3) Radius of tower B is 3 less than triple the radius of tower A: $r_B = 3r_A - 3$. **Step 1: Find the radius of tower A ($r_A$).** $\pi r_A^2 = 16.6 \Rightarrow r_A = \sqrt{\frac{16.6}{\pi}} \approx 2.3$ m. **Step 2: Find the radius of tower B ($r_B$).** $r_B = 3(2.3) - 3 = 6.9 - 3 = 3.9$ m. **Step 3: Calculate the height of tower A ($h_A$).** $120 = 2(16.6) + 2\pi(2.3)h_A$ $120 = 33.2 + 4.6\pi h_A$ $86.8 = 14.45 h_A \Rightarrow h_A \approx 6$ m. **Step 4: Calculate the height of tower B ($h_B$).** $120 = 2\pi(3.9)^2 + 2\pi(3.9)h_B$ $120 = 2\pi(15.21) + 7.8\pi h_B$ $120 = 95.55 + 24.5 h_B$ $24.45 = 24.5 h_B \Rightarrow h_B \approx 1$ m. **Answer:** The height of tower A is 6 m; the height of tower B is 1 m.