1. Problem statement: We want to write a vector function that describes the motion of the lollipop head based on the given data. 2. From the data, the position of the lollipop head at time $t$ is given by coordinates $(x, y)$. 3. A vector function for position can be written as: $$\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$$ where $x(t)$ and $y(t)$ are the position components in meters at time $t$ seconds. 4. Using the first few data points for $x$ and $y$: At $t=0.398$, $x=0.090908$, $y=0.917347$ At $t=0.432$, $x=0.101927$, $y=0.955914$ At $t=0.465$, $x=0.123966$, $y=1.019274$ 5. To find a functional form, we can approximate $x(t)$ and $y(t)$ by fitting a polynomial or other function. For simplicity, assume linear functions: $$x(t) = a_xt + b_x$$ $$y(t) = a_yt + b_y$$ 6. Using two points to find $a_x$ and $b_x$: From points $(0.398, 0.090908)$ and $(0.432, 0.101927)$: $$a_x = \frac{0.101927 - 0.090908}{0.432 - 0.398} = \frac{0.011019}{0.034} \approx 0.3241$$ $$b_x = 0.090908 - 0.3241 \times 0.398 \approx 0.090908 - 0.1289 = -0.038$$ 7. Similarly for $y(t)$ using points $(0.398, 0.917347)$ and $(0.432, 0.955914)$: $$a_y = \frac{0.955914 - 0.917347}{0.432 - 0.398} = \frac{0.038567}{0.034} \approx 1.1349$$ $$b_y = 0.917347 - 1.1349 \times 0.398 \approx 0.917347 - 0.451 = 0.4663$$ 8. Therefore, the vector function approximating the lollipop head's motion is: $$\vec{r}(t) = (0.3241t - 0.038)\hat{i} + (1.1349t + 0.4663)\hat{j}$$ 9. This function gives the position of the lollipop head at any time $t$ in seconds. 10. Note: For more accuracy, higher degree polynomials or spline fits can be used, but this linear approximation captures the general trend. Final answer: $$\boxed{\vec{r}(t) = (0.3241t - 0.038)\hat{i} + (1.1349t + 0.4663)\hat{j}}$$