What is the capacity of a pail in litres if it's in the shape of a frustum with top diameter 32cm, bottom diameter 24cm, and height 30cm? Also, calculate the determinant of the matrix A = [[3, -1], [4

Here are the solutions to the problems. 1. Capacity of the pail in litres Step 1: Identify the given dimensions of the frustum. Top diameter = 32 cm, so top radius R = (32)/(2) = 16 cm. Bottom diameter = 24 cm, so bottom radius r = (24)/(2) = 12 cm. Height h = 30 cm. Step 2: Calculate the volume of the frustum. The formula for the volume of a frustum is V = (1)/(3) h (R^2 + Rr + r^2). Using ≈ 3.14159: V = (1)/(3) × 3.14159 × 30 × (16^2 + 16 × 12 + 12^2) V = 10 × 3.14159 × (256 + 192 + 144) V = 31.4159 × (592) V = 18595.0048 cm^3 Step 3: Convert the volume from cubic centimeters to litres. Since 1 litre = 1000 cm^3: V_litres = (18595.0048)/(1000) V_litres = 18.5950048 litres Step 4: Round the answer. Rounding to two decimal places, the capacity is 18.60 litres. The capacity of the pail is 18.60 litres. 2. Calculate the determinant of the matrix A = 3 & -1 \\ 4 & 5 Step 1: Recall the formula for the determinant of a 2 × 2 matrix. For a matrix a & b \\ c & d , the determinant is ad - bc. Step 2: Substitute the values from matrix A. Here, a=3, b=-1, c=4, d=5. (A) = (3 × 5) - (-1 × 4) (A) = 15 - (-4) (A) = 15 + 4 (A) = 19 The determinant of matrix A is 19. 3. Without using table or a calculator Simplify each leaving your answer with rational denominators a) 45^ 60^ Step 1: Write down the exact values of 45^ and 60^. 45^ = sqrt(2)2 60^ = sqrt(3) Step 2: Multiply the values. 45^ 60^ = sqrt(2)2 × sqrt(3) = sqrt(2) × sqrt(3)2 = sqrt(6)2 The simplified expression is sqrt(6)2. b) 60^ 30^ + 60^ 30^ Step 1: Write down the exact values of each trigonometric function. 60^ = (1)/(2) 30^ = (1)/(2) 60^ = sqrt(3)2 30^ = sqrt(3)2 Step 2: Substitute the values into the expression. 60^ 30^ + 60^ 30^ = ((1)/(2) × (1)/(2)) + (sqrt(3)2 × sqrt(3)2) Step 3: Perform the multiplications. = (1)/(4) + (3)/(4) Step 4: Add the fractions. = (1+3)/(4) = (4)/(4) = 1 The simplified expression is 1. 3 done, 2 left today. You're making progress.