This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back Esete — missed you this week. Here are the solutions to the problems: 1. Find the total surface area of a right circular cone of radius 6 cm and height 12 cm. Step 1: Calculate the slant height (l) using the Pythagorean theorem. l = sqrt(r^2 + h^2) l = sqrt((6 cm))^2 + (12 cm)^2 l = sqrt(36 cm)^2 + 144 cm^2 l = sqrt(180 cm)^2 l = sqrt(36 × 5 cm)^2 l = 6sqrt(5) cm Step 2: Calculate the total surface area (TSA) of the cone. The formula for the total surface area of a cone is TSA = r (r + l). TSA = (6 cm) (6 cm + 6sqrt(5) cm) TSA = 6 (6 + 6sqrt(5)) cm^2 TSA = 36 (1 + sqrt(5)) cm^2 Using sqrt(5) ≈ 2.236: TSA ≈ 36 (1 + 2.236) cm^2 TSA ≈ 36 (3.236) cm^2 TSA ≈ 116.496 cm^2 TSA ≈ 116.496 × 3.14159 cm^2 TSA ≈ 366.0 cm^2 The total surface area is 36(1 + sqrt(5)) cm^2 or approximately 366.0 cm^2. 2. The area of the total surface of a right circular cone is 64 m^2 and its slant height is 5 times the radius of the base. Find the radius of the base. Step 1: Write down the formula for the total surface area (TSA) and the given information. TSA = r (r + l) Given TSA = 64 m^2 and l = 5r. Step 2: Substitute the given values and relationship into the formula. 64 m^2 = r (r + 5r) 64 m^2 = r (6r) 64 m^2 = 6 r^2 Step 3: Solve the equation for r. r^2 = (64)/(6) m^2 r^2 = (32)/(3) m^2 r = sqrt((32)/(3)) m r = sqrt(32)sqrt(3) m r = 4sqrt(2)sqrt(3) m To rationalize the denominator: r = 4sqrt(2)sqrt(3) × sqrt(3)sqrt(3) m r = 4sqrt(6)3 m Using ≈ 3.14159: r ≈ sqrt((32)/(3 × 3.14159)) m r ≈ sqrt((32)/(9.42477)) m r ≈ sqrt(3.3953) m r ≈ 1.8439 m The radius of the base is 4sqrt(6)3 m or approximately 1.84 m. Send me the next one 📸