91Ó°ÊÓ

The boat's actual speed is given as \(\sqrt{3} \mathrm{m} / \mathrm{s}\) and its direction is \(30^{\circ}\) north of east. We can find the eastward and northward components of the boat's actual speed using the cosine and sine functions, respectively. The eastward component is given by: \(v_x = v \cos (\theta)\) \(v_x = \sqrt{3} \mathrm{m} / \mathrm{s} \times \cos (30^{\circ})\) The northward component is given by: \(v_y = v \sin (\theta)\) \(v_y = \sqrt{3} \mathrm{m} / \mathrm{s} \times \sin (30^{\circ})\)

As the boat is in a current that flows eastward at a speed of 1 m/s, we simply subtract the eastward component of the current from the eastward component of the boat's speed to obtain the eastward component of the wind's velocity, \(w_x\). \(w_x = v_x - 1\mathrm{m} / \mathrm{s}\) For the northward component of the wind, we have no opposing velocity, so the value is simply equal to the northward component of the boat's actual speed. \(w_y = v_y\)

To find the wind's speed, we use the Pythagorean theorem. \(w = \sqrt{w_x^2 + w_y^2}\) To find the direction of the wind, we use arctangent to find the angle relative to the eastward direction. \(\beta = \arctan \left(\frac{w_y}{w_x}\right)\)

From Step 1, we have: \(v_x = \sqrt{3} \times \cos(30^{\circ}) = \sqrt{3} \times \frac{\sqrt{3}}{2} = \frac{3}{2} \mathrm{m}/\mathrm{s}\) \(v_y = \sqrt{3} \times \sin(30^{\circ}) = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \mathrm{m}/\mathrm{s}\) From Step 2, we have: \(w_x = \frac{3}{2} \mathrm{m}/\mathrm{s} - 1\mathrm{m}/\mathrm{s} =\frac{1}{2}\mathrm{m}/\mathrm{s}\) \(w_y = \frac{\sqrt{3}}{2} \mathrm{f}/\mathrm{s}\) Now, Step 3: \(w = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \mathrm{m} / \mathrm{s}\) And for the direction: \(\beta = \arctan \left(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\right) = \arctan (\sqrt{3}) = 60^{\circ}\) Hence, the wind's speed is \(1 \mathrm{m} / \mathrm{s}\), and its direction is \(60^{\circ}\) north of east.