**Problem 4: Spaceship Navigation**The position function of a spaceship is given by:\[ \mathbf{r}(t) = (3 + t)\mathbf{i} + (2 + \ln t)\mathbf{j} + \left(7 - \frac{4}{t^2 + 1}\right)\mathbf{k} \]The coordinates of a space station are \((6, 4, 9)\). The spaceship needs to coast into the space station without any further propulsion. The question is: **When should the engines be turned off?** To solve this, we need to find the time \( t \) at which the coordinates of the spaceship \( \mathbf{r}(t) \) correspond exactly to the coordinates of the space station \((6, 4, 9)\).1. Set the \( i \)-component of the position vector equal to the x-coordinate of the space station: \[ 3 + t = 6 \]2. Set the \( j \)-component of the position vector equal to the y-coordinate of the space station: \[ 2 + \ln t = 4 \]3. Set the \( k \)-component of the position vector equal to the z-coordinate of the space station: \[ 7 - \frac{4}{t^2 + 1} = 9 \]Solving these equations will give the time \( t \) at which the spaceship is at the coordinates of the space station.

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4. The position function of a spaceship is 4 r(t) = (3+ t)i + (2+ In t)j+ k t2 +1 7 - and the coordinates of a space station are (6, 4, 9). The captain needs the spaceship to coast into the space station. When should the engines be turned off?

4. The position function of a spaceship is 4 r(t) = (3+ t)i + (2+ In t)j+ k t2 +1 7 - and the coordinates of a space station are (6, 4, 9). The captain needs the spaceship to coast into the space station. When should the engines be turned off?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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