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Chapter 10: Problem 5
What is the polar equation of the vertical line \(x=5 ?\)
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Sector of a hyperbola Let \(H\) be the right branch of the hyperbola \(x^{2}-y^{2}=1\) and let \(\ell\) be the line \(y=m(x-2)\) that passes through the point (2,0) with slope \(m,\) where \(-\infty < m < \infty\). Let \(R\) be the region in the first quadrant bounded by \(H\) and \(\ell\) (see figure). Let \(A(m)\) be the area of \(R .\) Note that for some values of \(m\) \(A(m)\) is not defined. a. Find the \(x\) -coordinates of the intersection points between \(H\) and \(\ell\) as functions of \(m ;\) call them \(u(m)\) and \(v(m),\) where \(v(m) > u(m) > 1 .\) For what values of \(m\) are there two intersection points? b. Evaluate \(\lim _{m \rightarrow 1^{+}} u(m)\) and \(\lim _{m \rightarrow 1^{+}} v(m)\) c. Evaluate \(\lim _{m \rightarrow \infty} u(m)\) and \(\lim _{m \rightarrow \infty} v(m)\) d. Evaluate and interpret \(\lim _{m \rightarrow \infty} A(m)\)
Without using a graphing utility, determine the symmetries (if any) of the curve \(r=4-\sin (\theta / 2)\).
Eliminate the parameter to express the following parametric equations as a single equation in \(x\) and \(y.\) $$x=\tan t, y=\sec ^{2} t-1$$
A plane traveling horizontally at \(100 \mathrm{m} / \mathrm{s}\) over flat ground at an elevation of \(4000 \mathrm{m}\) must drop an emergency packet on a target on the ground. The trajectory of the packet is given by $$x=100 t, \quad y=-4.9 t^{2}+4000, \quad \text { for } t \geq 0,$$ where the origin is the point on the ground directly beneath the plane at the moment of the release. How many horizontal meters before the target should the packet be released in order to hit the target?
A plane traveling horizontally at \(80 \mathrm{m} / \mathrm{s}\) over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by $$x=80 t, \quad y=-4.9 t^{2}+3000, \quad \text { for } t \geq 0$$ where the origin is the point on the ground directly beneath the plane at the moment of the release. Graph the trajectory of the packet and find the coordinates of the point where the packet lands.
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