19. Convert the following equations from polar to rectangular form. \( \begin{array}{lll}\text { a. } 2 \sin (\theta)+3 \cos (\theta)=\mathrm{r} & \text { b. } r=3 \csc (\theta) & \text { c. } r^{2} \sin (2 \theta)=2\end{array} \)

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.

Follow the seven step strategy to graph the following rational function. \( f(x)=\frac{6 x}{x-2} \) To graph the function, first determine the symmetry of the graph of f . Choose the correct answer below. y-axis symmetry origin symmetry neither \( y \)-axis symmetry nor origin symmetry What is the \( y \)-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( y \)-intercept is (Type an integer or a simplified fraction.)

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 \( (-3,0) \)

A farmer decides to enclose a rectangular garden using the side of a barn as one side of the rectangle. What are the maximum dimensions that can be enclosed with 80 ft of fence? What is the maximum area of this garden? Select the correct choice below and fill in the answer box(s) to complete your choice. A. The area is maximized when both sides are \( \square \mathrm{nt} \). The area is maximized when the longer side is \( \square \mathrm{ft} \) and the shorter side is The maximum area that the farmer can enclose with 80 ft of fence is \( \square \mathrm{sq} \mathrm{ft} \)