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Chapter 6: Problem 46

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=10-0.01 e^{t}\\\ &y=0.4 t^{2} \end{aligned}$$

Short Answer

Expert verified
The curve represented by the provided parametric equations is graphed based on the points derived from inputting values of t in both functions for x and y. These points traced a curve when plotted, representing the behavior of the system.

Step by step solution

01

Set up plot values

Select a range of values for the parameter t. Normally, for simplicity, use t ranging from -10 to 10. These values usually capture enough variation in the graph to clearly see the entire shape of the graph. Utilize these t values to calculate corresponding x and y coordinates using provided equations.
02

Calculate for x and y

For each value of t from the selected range, substitute t into the given equations to get the x and y coordinates. \(x=10-0.01e^{t}\) and \(y=0.4t^{2}\). Calculate the x and y coordinates and pair them together to get points all over the graph, with each pair corresponding to a unique point on the curve.
03

Plotting the curve

Now we have a series of (x, y) pairs, each calculated from a certain value of t. Plug these coordinates into the graphing utility. Plot these points to graph the curve, with each pair of coordinate values representing a specific location on the x and y axis.
04

Noticing the Pattern

Keep the curve formed by the plotted points continuous as it represents the path traced by (x(t), y(t)) as t increases. Look at the formed pattern by the points on the graph to see if any curves are recognizable to interpret the behavior of the system that the parametric equations represent.

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Most popular questions from this chapter

The points represent the vertices of a triangle. (a) Draw triangle \(A B C\) in the coordinate plane, (b) find the altitude from vertex \(B\) of the triangle to side \(A C,\) and \((\mathrm{c})\) find the area of the triangle. $$A(1,1), B(2,4), C(3,5)$$

Find the distance between the parallel lines. (Graph can't copy) $$\begin{aligned} &3 x-4 y=1\\\ &3 x-4 y=10 \end{aligned}$$

A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\theta\) with the horizontal. The initial velocity is \(v_{0}\) feet per second, and the path of the projectile is modeled by the parametric equations $$x=\left(v_{0} \cos \theta\right) t$$ and $$y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}.$$ Use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\boldsymbol{\theta}\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and the range of the projectile. (a) \(\theta=60^{\circ}, \quad v_{0}=88\) feet per second (b) \(\theta=60^{\circ}, \quad v_{0}=132\) feet per second (c) \(\theta=45^{\circ}, \quad v_{0}=88\) feet per second (d) \(\theta=45^{\circ}, \quad v_{0}=132\) feet per second

Determine whether the statement is true or false. Justify your answer. A line that has an inclination greater than \(\pi / 2\) radians has a negative slope.

Consider the path of a projectile projected horizontally with a velocity of \(v\) feet per second at a height of \(s\) feet, where the model for the path is $$x^{2}=-\frac{v^{2}}{16}(y-s)$$ In this model (in which air resistance is disregarded), \(y\) is the height (in feet) of the projectile and \(x\) is the horizontal distance (in feet) the projectile travels. A ball is thrown from the top of a 100 -foot tower with a velocity of 28 feet per second. A. Find the equation of the parabolic path. B. How far does the ball travel horizontally before striking the ground?
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