Directions: In order to receive credit for this problem, you must solve it by following the steps indicated. Failure to do so will result in no credit. On his way to campus, Jim decides to pick up a dozen donuts, some of which he hopes will survive the trip from the donut shop to his ofce. Since Jim plans to to make the same trip again (and again), he wants to gure out where he should park as to minimize the distance he must walk from his car to the donut shop. A diagram is shown below of the road and the donut shop, which is located at (2,4). Two points on the road, (0, 1) and (4, 3), are also shown on the image below. Jim can park anywhere on the road that he would like. To minimize the distance he has to walk, Jim needs to nd the shortest distance from the road, which is modeled by a line, to the point where the donut shop is located. He decides to perform this computation as follows. A. [3 pts] Show that the vector-valued function E (t) 2 (4t, 2t + 1) parameterizes the line in the above gure. B. [3 pts] Suppose that Jim decides to perk his car at the point (4, 3) on the road. i. [1 pt] 0n the gure below, sketch 3 vector that begins at (4, 3) and ends at the donut shop. ii. [2 pts] Explain why (4, 3) is not the point on the line that is closest to the donut shop. Hint: Draw a certain triangle on the image, where the vector you sketched is directed along one of the sides. y C. [3 pts] We can make a new vector-valued function d (t) that extends from any point (1r(t),y(t)) on the line to the donut shop. i. [1 pt] On the image below, label a point (52(t), y(t)) on the line, write 9:05) and y(t) in terms of t, then draw the vector d (t) that extends from that point to the donut shop. ii. [2 pts] Show that d(t) = (2 415,3 2t). Hint: You have the point where d(t) starts in terms of t and the point where it ends. How do you nd a vector that originates at one point and extends to another? 3; Donut shop . . . (2: 4) _ D. [6 pts] The magnitude [(1 (t)[ is the distance between the point and the line. Find the value of if so [(1 (t)[ is minimized, then use it to nd the minimum distance [d (t)[ between the line and the donut shop. Make sure to justify your work. Hint: You can proceed in many ways. One useful way is to consider the orientation of d(t) and a direction vector for the line. E. [2 pts] Fill in the bubble that correctly completes the statement below. The scenario described in this problem requires that we choose a parameterization of the line that models the road. However, the distance between the road and the donut shop depend on how we parameterize the line. should O should not F. [8 pts] To explore your response to Part E, do the following. Find a different parameterization for the line in this problem. Use this parameterization and mimic the earlier procedure to calculate the distance between the line. Does your answer match your predicted response in Part E? Donut shop (2, 4) 3 (4, 3) Road (0, 1) T 1 1 2 3 4 5