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.
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You're on a roll — here is the solution to the questions: Question 2.2: Steel needed for the gate Step 1: Convert the breadth to meters. The height of the gate is 1.2 m. The breadth of the gate is 95 cm. 95 cm = 95 × 1 m100 cm = 0.95 m Step 2: Calculate the total length of the sides (perimeter) of the gate. The gate is rectangular, so it has two sides of 1.2 m and two sides of 0.95 m. Length of sides = 2 × (height + breadth) Length of sides = 2 × (1.2 m + 0.95 m) Length of sides = 2 × 2.15 m Length of sides = 4.3 m Step 3: Calculate the length of one diagonal using the Pythagorean Theorem. The diagonals form right-angled triangles with the sides of the gate. Let d be the length of a diagonal. d^2 = (height)^2 + (breadth)^2 d^2 = (1.2 m)^2 + (0.95 m)^2 d^2 = 1.44 m^2 + 0.9025 m^2 d^2 = 2.3425 m^2 d = sqrt(2.3425) m d ≈ 1.5305 m Step 4: Calculate the total length of the two diagonals. Since there are two diagonals of equal length: Length of diagonals = 2 × d Length of diagonals = 2 × 1.5305 m Length of diagonals ≈ 3.061 m Step 5: Calculate the total amount of steel needed. Total steel = Length of sides + Length of diagonals Total steel = 4.3 m + 3.061 m Total steel = 7.361 m Question 2.3: Water Tank and Cylindrical Buckets 2.3.1 Calculate the volume of the rectangular tank. Step 1: Identify the dimensions of the rectangular tank from the diagram. Length L = 6 m Breadth B = 5 m Height H = 5 m Step 2: Apply the formula for the volume of a rectangular prism. V_tank = L × B × H V_tank = 6 m × 5 m × 5 m V_tank = 150 m^3 2.3.2 Calculate the volume of the cylindrical bucket. Round off to three decimal places. Step 1: Identify the dimensions of the cylindrical bucket from the diagram and given . Diameter D = 0.5 m, so radius r = (D)/(2) = 0.5 m2 = 0.25 m Height h = 1 m = 3.14 Step 2: Apply the formula for the volume of a cylinder. V_bucket = r^2 h V_bucket = 3.14 × (0.25 m)^2 × 1 m V_bucket = 3.14 × 0.0625 m^2 × 1 m V_bucket = 0.19625 m^3 Step 3: Round the volume to three decimal places. V_bucket = 0.196 m^3 2.3.3 How many full cylindrical buckets will be needed to fill the rectangular tank to 75% capacity? Step 1: Calculate 75% capacity of the rectangular tank. 75% capacity = 0.75 × V_tank 75% capacity = 0.75 × 150 m^3 75% capacity = 112.5 m^3 Step 2: Calculate the number of buckets needed. Use the unrounded volume of the bucket for accuracy. Number of buckets = 75% capacity of tankV_bucket Number of buckets = 112.5 m^30.19625 m^3 Number of buckets ≈ 573.78 Step 3: Round up to the nearest whole number since only full buckets can be used. Number of buckets = 574 buckets Drop the next question.