A) Calculate the length of BC. b) Calculate the value of angle BAC in degrees, correct to 2 decimal places using tangent tables only. 40. Determine the value of x.

$$13^2 + BC^2 = 85^2$$ $$169 + BC^2 = 7225$$

$$BC^2 = 7225 - 169$$ $$BC^2 = 7056$$

$$BC = \sqrt{7056}$$ $$BC = 84 \text{ cm}$$

The length of BC is $\boxed{\text{84 cm}}$.

b) Calculate the value of angle BAC in degrees, correct to 2 decimal places using tangent tables only.

$$\tan(\angle BAC) = \frac{84}{13}$$ $$\tan(\angle BAC) \approx 6.461538$$

$$\angle BAC = \arctan(6.461538)$$ $$\angle BAC \approx 81.2006^\circ$$

$$\angle BAC \approx 81.20^\circ$$

The value of angle BAC is $\boxed{81.20^\circ}$.

40.
Determine the value of $x$.

Given $\cos(3x^\circ) = \sin((x-60)^\circ)$, we can rewrite $\cos(3x^\circ)$ as $\sin(90^\circ - 3x^\circ)$.
So, the equation becomes:

$$\sin(90^\circ - 3x^\circ) = \sin((x-60)^\circ)$$

$$90 + 60 = x + 3x$$ $$150 = 4x$$ $$x = \frac{150}{4}$$ $$x = 37.5$$

The value of $x$ is $\boxed{37.5}$.

41.

a) Calculate the amount of money in Ksh, that she obtained after exchanging the foreign currencies.

The bank buys foreign currency.

$$\text{Ksh from Sterling Pounds} = 1580 \text{ Sterling Pounds} \times 152 \text{ Ksh/Sterling Pound}$$ $$\text{Ksh from Sterling Pounds} = 239920 \text{ Ksh}$$

The bank buys foreign currency.

$$\text{Ksh from Euros} = 1205 \text{ Euros} \times 125 \text{ Ksh/Euro}$$ $$\text{Ksh from Euros} = 150625 \text{ Ksh}$$

$$\text{Total Ksh} = 239920 \text{ Ksh} + 150625 \text{ Ksh}$$ $$\text{Total Ksh} = 390545 \text{ Ksh}$$

The businesswoman obtained $\boxed{390545 \text{ Ksh}}$.

b) Calculate the amount of money, in Ksh that she needed for the exchange.

The bank sells foreign currency.

$$\text{Ksh for Euros} = 1250 \text{ Euros} \times 128 \text{ Ksh/Euro}$$ $$\text{Ksh for Euros} = 160000 \text{ Ksh}$$

The bank sells foreign currency.

$$\text{Ksh for Sterling Pounds} = 1520 \text{ Sterling Pounds} \times 158 \text{ Ksh/Sterling Pound}$$ $$\text{Ksh for Sterling Pounds} = 240160 \text{ Ksh}$$

$$\text{Total Ksh needed} = 160000 \text{ Ksh} + 240160 \text{ Ksh}$$ $$\text{Total Ksh needed} = 400160 \text{ Ksh}$$

The trader needed $\boxed{400160 \text{ Ksh}}$.