1. **State the problem:** A rigid body rotates at 5 revolutions per second about an axis through the origin with direction ratios 2, -1, 3. We need to find the magnitude of the velocity of the body at the point (2, 4, 3). 2. **Convert angular velocity to radians per second:** Since 1 revolution = $2\pi$ radians, angular velocity $\omega = 5 \times 2\pi = 10\pi$ radians/second. 3. **Find the unit vector along the axis of rotation:** Direction ratios are (2, -1, 3). The magnitude is $$\sqrt{2^2 + (-1)^2 + 3^2} = \sqrt{4 + 1 + 9} = \sqrt{14}.$$ Unit vector $\hat{n} = \left(\frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}}, \frac{3}{\sqrt{14}}\right)$. 4. **Find the position vector of the point:** $\vec{r} = (2, 4, 3)$. 5. **Find the component of $\vec{r}$ along the axis:** Projection of $\vec{r}$ on $\hat{n}$ is $$r_{\parallel} = \vec{r} \cdot \hat{n} = 2 \times \frac{2}{\sqrt{14}} + 4 \times \frac{-1}{\sqrt{14}} + 3 \times \frac{3}{\sqrt{14}} = \frac{4 - 4 + 9}{\sqrt{14}} = \frac{9}{\sqrt{14}}.$$ 6. **Find the perpendicular component of $\vec{r}$ to the axis:** $$\vec{r}_{\perp} = \vec{r} - r_{\parallel} \hat{n} = (2,4,3) - \frac{9}{\sqrt{14}} \left(\frac{2}{\sqrt{14}}, \frac{-1}{\sqrt{14}}, \frac{3}{\sqrt{14}}\right) = (2,4,3) - \frac{9}{14} (2, -1, 3).$$ Calculate: $$\frac{9}{14} \times 2 = \frac{18}{14} = \frac{9}{7}, \quad \frac{9}{14} \times (-1) = -\frac{9}{14}, \quad \frac{9}{14} \times 3 = \frac{27}{14}.$$ So, $$\vec{r}_{\perp} = \left(2 - \frac{9}{7}, 4 - \left(-\frac{9}{14}\right), 3 - \frac{27}{14}\right) = \left(\frac{14}{7} - \frac{9}{7}, \frac{56}{14} + \frac{9}{14}, \frac{42}{14} - \frac{27}{14}\right) = \left(\frac{5}{7}, \frac{65}{14}, \frac{15}{14}\right).$$ 7. **Calculate the magnitude of $\vec{r}_{\perp}$:** $$\left|\vec{r}_{\perp}\right| = \sqrt{\left(\frac{5}{7}\right)^2 + \left(\frac{65}{14}\right)^2 + \left(\frac{15}{14}\right)^2} = \sqrt{\frac{25}{49} + \frac{4225}{196} + \frac{225}{196}}.$$ Convert $\frac{25}{49}$ to denominator 196: $$\frac{25}{49} = \frac{100}{196}.$$ Sum inside the root: $$\frac{100}{196} + \frac{4225}{196} + \frac{225}{196} = \frac{4550}{196} = \frac{4550}{196}.$$ Simplify: $$\frac{4550}{196} = \frac{4550 \div 2}{196 \div 2} = \frac{2275}{98}.$$ So, $$\left|\vec{r}_{\perp}\right| = \sqrt{\frac{2275}{98}} = \frac{\sqrt{2275}}{\sqrt{98}}.$$ Calculate approximate values: $\sqrt{2275} \approx 47.7$, $\sqrt{98} \approx 9.9$, so $$\left|\vec{r}_{\perp}\right| \approx \frac{47.7}{9.9} \approx 4.82.$$ 8. **Calculate the magnitude of velocity:** Velocity magnitude $v = \omega \times \left|\vec{r}_{\perp}\right| = 10\pi \times 4.82 \approx 10 \times 3.1416 \times 4.82 = 151.5$ units/second. **Final answer:** The magnitude of the velocity at point (2,4,3) is approximately $151.5$ units per second.