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Chapter 6: Problem 40

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=|t+2|\\\ &y=3-t \end{aligned}$$

Short Answer

Expert verified
The graph of the parametric equations is a pair of straight lines that meet at the point where \( t=-2 \), corresponding to the point (0,5), and then diverge. One line extends to the left and upwards, representing the situation when \( t < -2 \). Another line extends to the right and downwards, representing the situation when \( t \geq -2 \).

Step by step solution

01

Understand Parametric Equations

In a parametric equation, both x and y are expressed in terms of a third variable, known as the parameter. In this case, the parameter is \( t \). Thus, we can view the equations \( x=|t+2| \) and \( y=3-t \) as instructions for how x and y change as \( t \) varies.
02

Interpret the absolute value function on x

The absolute value function \( |t+2| \) will be greater than or equal to zero regardless of the value of \( t \). If \( t \) is less than -2, then \( t+2 \) is negative, so the absolute value of \( t+2 \) is \( 2-t \). If \( t \) is greater than or equal to -2, then \( t+2 \) is non-negative, so the absolute value of \( t+2 \) is \( t+2 \). Hence we have the piecewise function: \( x=2-t \) if \( t<-2 \), and \( x=t+2 \) if \( t\geq-2 \).
03

Understand the linear equation on y

The second equation \( y=3-t \) is a linear equation, so y decreases linearly as \( t \) increases.
04

Graph the parametric equations

Using a graphing utility, we plot the respective x and y values for a range of \( t \) values using the interpretation from step 2 for \( x \) and step 3 for \( y \). We should see two distinct straight lines formed, meeting at the point when \( t = -2 \).
05

Interpret the graph

The graph starts at the point where \( t \) is very negative (leftmost point of the graph), and as \( t \) increases, the graph moves from left to right. The straight line portion where \( t<-2 \) corresponds to \( x=2-t \), and the portion where \( t\geq-2 \) corresponds to \( x=t+2 \). The y values decrease linearly across the entire graph.

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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

An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an angle of \(15^{\circ}\) with the horizontal and at an initial speed of 225 feet per second. (a) Write a set of parametric equations that model the path of the arrow. (See Exercises 93 and 94 .) (b) Assuming the ground is level, find the distance the arrow travels before it hits the ground. (Ignore air resistance.) (c) Use a graphing utility to graph the path of the arrow and approximate its maximum height. (d) Find the total time the arrow is in the air.
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