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Chapter 6: Problem 17
Identify the conic and sketch its graph. \(r=\frac{5}{1+\sin \theta}\)
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Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(\frac{x^{2}}{64}+\frac{y^{2}}{28}=1\)
Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,卤8)\(;\) foci: (0,卤4)
The relation between the velocity \(y\) (in radians per second) of a pendulum and its angular displacement \(\theta\) from the vertical can be modeled by a semiellipse. A 12-centimeter pendulum crests \((y=0)\) when the angular displacement is -0.2 radian and 0.2 radian. When the pendulum is at equilibrium \((\theta=0),\) the velocity is -1.6 radians per second. (a) Find an equation that models the motion of the pendulum. Place the center at the origin. (b) Graph the equation from part (a). (c) Which half of the ellipse models the motion of the pendulum?
Use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for \(y\) and obtain two equations.) \(12 x^{2}+20 y^{2}-12 x+40 y-37=0\)
Find the standard form of the equation of the parabola with the given characteristics. Vertex: (1,2)\(;\) directrix: \(y=-1\)
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