Araba walked $2t$ km from village S to village T on a bearing of $065^\circ$. Then, she walked $3t$ km from village T to town U on a bearing of $155^\circ$. The distance between S and U is $6\sqrt{13}$ km. We need to: (a) Illustrate the information in a diagram. (b) Calculate the value of $t$ to the nearest whole number. (c) Calculate the bearing of U from S.

1. Problem Description

Araba walked $2t$ km from village S to village T on a bearing of $065^\circ$. Then, she walked $3t$ km from village T to town U on a bearing of $155^\circ$. The distance between S and U is $6\sqrt{13}$ km. We need to:

(a) Illustrate the information in a diagram.

(b) Calculate the value of $t$ to the nearest whole number.

(c) Calculate the bearing of U from S.

2. Solution Steps

(a) Diagram:

The diagram consists of three points, S, T, and U.

- From S, draw a line ST of length $2t$ km at a bearing of $065^\circ$.

- From T, draw a line TU of length $3t$ km at a bearing of $155^\circ$.

- Connect S and U with a line of length $6\sqrt{13}$ km.

- Let $\angle NST = 65^\circ$.

- Let $\angle STE = 155^\circ$, where E is a point such that TE extends the line ST. Then $\angle ETS = 155^\circ$. The angle between the north direction at T and TU is $155^\circ$. Let the bearing of U from T be $155^\circ$.

- The angle $\angle STU$ is obtained by $180^\circ - (155^\circ - 65^\circ)= 180^\circ - 90^\circ = 90^\circ$.

(b) Value of $t$:

Since $\angle STU = 90^\circ$, the triangle STU is a right-angled triangle. We can use the Pythagorean theorem:

$SU^2 = ST^2 + TU^2$

$(6\sqrt{13})^2 = (2t)^2 + (3t)^2$

$36 \times 13 = 4t^2 + 9t^2$

$468 = 13t^2$

$t^2 = \frac{468}{13} = 36$

$t = \sqrt{36} = 6$

(c) Bearing of U from S:

Let $\theta = \angle TSU$. Then, we have

$\tan \theta = \frac{TU}{ST} = \frac{3t}{2t} = \frac{3}{2} = 1.5$

$\theta = \arctan(1.5) \approx 56.31^\circ$

The bearing of T from S is $065^\circ$. The bearing of U from S is $65^\circ + \theta$.

Bearing of U from S = $65^\circ + 56.31^\circ = 121.31^\circ$

Rounding to the nearest whole number, the bearing of U from S is $121^\circ$.

3. Final Answer

(a) The diagram is as described above.

(b) (i) $t = 6$

(ii) Bearing of U from S = $121^\circ$