Step 1: Calculate the length of BC using the Pythagorean theorem. Given a right-angled triangle ABC, right-angled at B: $AB = 13 \text{ cm}$ $AC = 85 \text{ cm}$ The Pythagorean theorem states $AC^2 = AB^2 + BC^2$. $$85^2 = 13^2 + BC^2$$ $$7225 = 169 + BC^2$$ $$BC^2 = 7225 - 169$$ $$BC^2 = 7056$$ $$BC = \sqrt{7056}$$ $$BC = 84 \text{ cm}$$ The length of BC is $\boxed{\text{84 cm}}$. Step 2: Calculate the value of angle BAC. Let $\theta = \angle BAC$. We use the tangent ratio: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{BC}{AB}$. $$\tan(\theta) = \frac{84}{13}$$ $$\tan(\theta) \approx 6.461538$$ To find $\theta$, we take the inverse tangent: $$\theta = \arctan(6.461538)$$ $$\theta \approx 81.2009^\circ$$ Correct to 2 decimal places, the value of angle BAC is $\boxed{81.20^\circ}$. Step 1: Use the trigonometric identity $\cos A = \sin(90^\circ - A)$. Given the equation $\cos(3x^\circ) = \sin(x-60^\circ)$. We can rewrite $\cos(3x^\circ)$ as $\sin(90^\circ - 3x^\circ)$. So, the equation becomes: $$\sin(90^\circ - 3x^\circ) = \sin(x-60^\circ)$$ Step 2: Solve for $x$. For $\sin A = \sin B$, the general solutions are $A = B + 360^\circ k$ or $A = 180^\circ - B + 360^\circ k$, where $k$ is an integer. Since $x$ is an acute angle ($0^\circ < x < 90^\circ$), we consider the principal values. Case 1: $90^\circ - 3x = x - 60^\circ$ $$90^\circ + 60^\circ = x + 3x$$ $$150^\circ = 4x$$ $$x = \frac{150^\circ}{4}$$ $$x = 37.5^\circ$$ This value of $x$ is acute. Let's verify: $\cos(3 \times 37.5^\circ) = \cos(112.5^\circ)$ $\sin(37.5^\circ - 60^\circ) = \sin(-22.5^\circ)$ Since $\cos(112.5^\circ) = -\sin(22.5^\circ)$ and $\sin(-22.5^\circ) = -\sin(22.5^\circ)$, the equality holds. Case 2: $90^\circ - 3x = 180^\circ - (x - 60^\circ)$ $$90^\circ - 3x = 180^\circ - x + 60^\circ$$ $$90^\circ - 3x = 240^\circ - x$$ $$90^\circ - 240^\circ = 3x - x$$ $$-150^\circ = 2x$$ $$x = -75^\circ$$ This value of $x$ is not acute, so it is rejected. The value of $x$ is $\boxed{37.5^\circ}$. a) Calculate the amount of money in Ksh, that she obtained after exchanging the foreign currencies. Step 1: Determine the exchange rates for the businesswoman. The businesswoman arrived in Kenya with foreign currency (Sterling Pounds and Euros) and wants to exchange them for Kenyan Shillings (Ksh). The bank will buy her foreign currency. • Bank buying rate for 1 Euro = 125 Ksh • Bank buying rate for 1 Sterling Pound = 152 Ksh Step 2: Calculate the amount obtained from Sterling Pounds. Amount of Sterling Pounds = 1580 Amount in Ksh = $1580 \times 152 \text{ Ksh}$ $$1580 \times 152 = 239960 \text{ Ksh}$$ Step 3: Calculate the amount obtained from Euros. Amount of Euros = 1205 Amount in Ksh = $1205 \times 125 \text{ Ksh}$ $$1205 \times 125 = 150625 \text{ Ksh}$$ Step 4: Calculate the total amount obtained. Total amount = Amount from Sterling Pounds + Amount from Euros $$239960 \text{ Ksh} + 150625 \text{ Ksh} = 390585 \text{ Ksh}$$ The businesswoman obtained $\boxed{390585 \text{ Ksh}}$. b) Calculate the amount of money, in Ksh that she needed for the exchange. Step 1: Determine the exchange rates for the trader. The trader intends to travel abroad and needs foreign currency (Euros and Sterling Pounds). The bank will sell the foreign currency to the trader. • Bank selling rate for 1 Euro = 128 Ksh • Bank selling rate for 1 Sterling Pound = 158 Ksh Step 2: Calculate the cost for Euros. Amount of Euros needed = 1250 Cost in Ksh = $1250 \times 128 \text{ Ksh}$ $$1250 \times 128 = 160000 \text{ Ksh}$$ Step 3: Calculate the cost for Sterling Pounds. Amount of Sterling Pounds needed = 1520 Cost in Ksh = $1520 \times 158 \text{ Ksh}$ $$1520 \times 158 = 240160 \text{ Ksh}$$ Step 4: Calculate the total amount needed. Total amount needed = Cost for Euros + Cost for Sterling Pounds $$160000 \text{ Ksh} + 240160 \text{ Ksh} = 400160 \text{ Ksh}$$ The trader needed $\boxed{400160 \text{ Ksh}}$.