1. \( 25 x^{2}+y^{2}-200 x+6 y+384=0 \), centro 2. \( (x+1)^{2}+(y+8)^{2}=100 \), intersecciones con el eje \( x \) 3. \( y^{2}-(x-2)^{2}=1 \), intersecciones con el eje \( y \) 4. \( y^{2}-y+3 x=3 \), pendiente de la recta tangente en \( (1,1) \) 5. \( x=t^{3}, y=4 t^{3} \), nombre de la gráfica rectangular 6. \( x=t^{2}-1, y=t^{3}+t+1 \), intersecciones con el eje \( y \) 7. \( r=-2 \cos \theta \), nombre de la gráfica polar 8. \( r=2+\operatorname{sen} \theta \), nombre de la gráfica polar 19. \( r=\operatorname{sen} 3 \theta, \operatorname{tangentes~a~la~gráfica~en~el~origen~} \) 20. \( r=\frac{1}{2+5 \operatorname{sen} \theta} \), excentricidad 21. \( r=\frac{10}{1-\operatorname{sen} \theta} \), foco yértice 22. \( r=\frac{12}{2+\cos \theta} \), centro

The exponential model \( \mathrm{A}=474.2 e^{0.006 t} \) describes the population, A , of a country in millions, \( t \) years after 2003 . Use the model to determine the population of the country in 2003 . The population of the country in 2003 was \( \square \) million.

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci at \( (-7,0) \) and \( (7,0) \); vertices at \( (3,0) \) and \( (-3,0) \)

Suppose that the functions \( f \) and \( g \) are defined as follows. \[ \begin{array}{l}f(x)=\sqrt{5 x+3} \\ g(x)=3 x^{2}+1\end{array} \] Find \( f+g \) and \( f \cdot g \). Then, give their domains using interval notation.

What is the dot product of vector \( h \) with a magnitude of 5 and vector \( i \) with a magnitude of 8 if the angle between the vectors is \( \frac{\pi}{6} \) radians? (Round your answer to the nearest hundredth.)

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci at \( (0,-8) \) and \( (0,8) \); vertices at \( (0,3) \) and \( (0,-3) \)