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 $R = \frac{32}{2} = 16$ cm.
Bottom diameter $= 24$ cm, so bottom radius $r = \frac{24}{2} = 12$ cm.
Height $h = 30$ cm.

Step 2: Calculate the volume of the frustum.
The formula for the volume of a frustum is $V = \frac{1}{3}\pi h (R^2 + Rr + r^2)$.
Using $\pi \approx 3.14159$:

$$V = \frac{1}{3} \times 3.14159 \times 30 \times (16^2 + 16 \times 12 + 12^2)$$ $$V = 10 \times 3.14159 \times (256 + 192 + 144)$$ $$V = 31.4159 \times (592)$$ $$V = 18595.0048 \text{ cm}^3$$

Step 3: Convert the volume from cubic centimeters to litres.
Since $1 \text{ litre} = 1000 \text{ cm}^3$:

$$V_{\text{litres}} = \frac{18595.0048}{1000}$$ $$V_{\text{litres}} = 18.5950048 \text{ litres}$$

Step 4: Round the answer.
Rounding to two decimal places, the capacity is 18.60 litres.
The capacity of the pail is $\boxed{18.60 \text{ litres}}$.

2. Calculate the determinant of the matrix $A = \begin{bmatrix} 3 & -1 \\ 4 & 5 \end{bmatrix}$

Step 1: Recall the formula for the determinant of a $2 \times 2$ matrix.
For a matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$.

Step 2: Substitute the values from matrix $A$.
Here, $a=3$, $b=-1$, $c=4$, $d=5$.

$$\det(A) = (3 \times 5) - (-1 \times 4)$$ $$\det(A) = 15 - (-4)$$ $$\det(A) = 15 + 4$$ $$\det(A) = 19$$

The determinant of matrix $A$ is $\boxed{19}$.

3. Without using table or a calculator Simplify each leaving your answer with rational denominators