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Chapter 8: Problem 70
$$ \text { For Exercises 65-76, simplify and express in standard form. } $$ $$ (3-5 i)^{2} $$
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In Exercises 79 and 80, explain the mistake that is made. Convert the rectangular coordinate \((-\sqrt{3}, 1)\) to polar coordinates. Solution: Label \(x\) and \(y . \quad x=-\sqrt{3}, y=1\) Find \(r . \quad r=\sqrt{x^{2}+y^{2}}=\sqrt{3+1}=\sqrt{4}=2\) Find \(\theta\). $$ \begin{aligned} \tan \theta &=\frac{1}{-\sqrt{3}}=-\frac{1}{\sqrt{3}} \\ \theta &=\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)=-\frac{\pi}{4} \end{aligned} $$ Write the point in polar \(\quad\left(2,-\frac{\pi}{4}\right)\) coordinates. This is incorrect. What mistake was made?
For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Flight of a Projectile. A projectile is launched from the ground at a speed of 400 feet per second at an angle of \(45^{\circ}\) with the horizontal. After how many seconds does the projectile hit the ground?
In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=4 \cos (2 t), y=t, t \text { in }[0,2 \pi] $$
In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r=-2 \cos \theta $$
For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Bullet Fired. A gun is fired from the ground at an angle of \(60^{\circ}\), and the bullet has an initial speed of 700 feet per second. How high does the bullet go? What is the horizontal (ground) distance between where the gun was fired and where the bullet hit the ground?
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