Calculate the surface area of the hexagonal prism. The base is a regular hexagon of sides 10 cm, and the height is 30 cm.

You're on a roll — Here are the solutions to the questions: Question 4: Calculate the surface area of the hexagonal prism. The base is a regular hexagon of sides 10 cm, and the height is 30 cm. Step 1: Calculate the area of one equilateral triangle forming the base using Heron's formula. A regular hexagon can be divided into 6 equilateral triangles. For an equilateral triangle with side s = 10 cm: The semi-perimeter p is: p = 10 cm + 10 cm + 10 cm2 = 30 cm2 = 15 cm Using Heron's formula, the area of one equilateral triangle (A_triangle) is: A_triangle = sqrt(p(p-s)(p-s)(p-s)) A_triangle = sqrt(15(15-10)(15-10)(15-10)) A_triangle = sqrt(15 × 5 × 5 × 5) A_triangle = sqrt(15 × 125) A_triangle = sqrt(1875) A_triangle = sqrt(625 × 3) = 25sqrt(3) cm^2 Step 2: Calculate the area of the hexagonal base (A_base). Since the hexagon consists of 6 such equilateral triangles: A_base = 6 × A_triangle A_base = 6 × 25sqrt(3) cm^2 = 150sqrt(3) cm^2 Step 3: Calculate the lateral surface area (A_lateral). The lateral surface consists of 6 rectangular faces. Each face has a width equal to the side of the hexagon (10 cm) and a height equal to the prism's height (30 cm). Area of one rectangular face = 10 cm × 30 cm = 300 cm^2. A_lateral = 6 × 300 cm^2 = 1800 cm^2 Step 4: Calculate the total surface area (SA_total). The total surface area is the sum of the areas of the two bases and the lateral surface area. SA_total = 2 × A_base + A_lateral SA_total = 2 × 150sqrt(3) cm^2 + 1800 cm^2 SA_total = 300sqrt(3) cm^2 + 1800 cm^2 Using sqrt(3) ≈ 1.73205: SA_total ≈ 300 × 1.73205 cm^2 + 1800 cm^2 SA_total ≈ 519.615 cm^2 + 1800 cm^2 SA_total ≈ 2319.615 cm^2 Rounding to two decimal places: SA_total ≈ 2319.62 cm^2 2319.62 cm^2 Question 5: Calculate the volume of the cone to the nearest millimetre. The cone has a radius of 5 cm and a perpendicular height of 8 cm. Use = 3.14. Step 1: Identify the given values. Radius (r) = 5 cm Height (h) = 8 cm = 3.14 Step 2: Use the formula for the volume of a cone. The volume (V) of a cone is given by: V = (1)/(3) r^2 h Step 3: Substitute the values and calculate the volume in cubic centimetres. V = (1)/(3) × 3.14 × (5 cm)^2 × 8 cm V = (1)/(3) × 3.14 × 25 cm^2 × 8 cm V = (1)/(3) × 3.14 × 200 cm^3 V = (628)/(3) cm^3 V ≈ 209.3333... cm^3 Step 4: Convert the volume to cubic millimetres and round to the nearest whole number. Since 1 cm = 10 mm, then 1 cm^3 = (10 mm)^3 = 1000 mm^3. V ≈ 209.3333... × 1000 mm^3 V ≈ 209333.333... mm^3 Rounding to the nearest whole number (nearest cubic millimetre): V ≈ 209333 mm^3 209333 mm^3 Send me the next one 📸