Question

1.two friends, suban and jas, start from the same point but ride their bikes down two different paths. the paths diverge at an angle of 38°. suban rides down his path at 12 km/h, and jas rides down his path at 14 km/h. determine how far apart the two are after riding at their respective speeds for 90 min. 5t

Explanation:

Step1: Calculate the distances each has traveled

Since \(1\) hour \( = 60\) minutes, \(90\) minutes \(=\frac{90}{60}=1.5\) hours.
Using the formula \(d = vt\) (distance \(d\), velocity \(v\), time \(t\)), for Suban: \(d_1=12\times1.5 = 18\) km. For Jas: \(d_2 = 14\times1.5=21\) km.

Step2: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) between sides \(a\) and \(b\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\).
Let \(a = 18\), \(b = 21\), and \(\theta=38^{\circ}\). Then \(c^{2}=18^{2}+21^{2}-2\times18\times21\times\cos(38^{\circ})\).
First, calculate \(18^{2}=324\), \(21^{2} = 441\), and \(2\times18\times21=756\). \(\cos(38^{\circ})\approx0.788\).
So \(c^{2}=324 + 441-756\times0.788\).
\(c^{2}=324+441 - 595.728\).
\(c^{2}=765-595.728=169.272\).
Take the square - root: \(c=\sqrt{169.272}\approx13\) km.

Answer:

They are approximately \(13\) km apart.