Based on the following table, which shows crashworthiness ratings for several categories of motor vehicles. In all of these exercises, take X as the crash-test rating of a small car, Y as the crash-test rating for a small SUV, and so on, as shown in the table.

$$\begin{matrix} & \text{} & \text{} & \text{Overall Frontal Crash Test Rating}\\ & \text{Number Tested} & \text{3 (Good)} & \text{2 (Acceptable)} & \text{1 (Marginal)} & \text{0 (Poor)}\\ \text{Small Cars X} & \text{16} & \text{1} & \text{11} & \text{2} & \text{2}\\ \text{Small SUVs Y} & \text{10} & \text{1} & \text{4} & \text{4} & \text{1}\\ \text{Medium SUVs Z} & \text{15} & \text{3} & \text{5} & \text{3} & \text{4}\\ \text{Passenger Vans U} & \text{13} & \text{3} & \text{0} & \text{3} & \text{7}\\ \text{Midsize Cars V} & \text{15} & \text{3} & \text{5} & \text{0} & \text{7}\\ \text{Large Cars W} & \text{19} & \text{9} & \text{5} & \text{3} & \text{2}\\ \end{matrix}$$

You choose, at random, a small car and a small SUV. What is the probability that both will be rated at least 2?

A college dean plans a student survey to estimate the percentage of currently enrolled students who plan to take classes during the next summer session. The desired margin of error for a 95% confidence interval is to be at most 5%. What is the necessary sample size?

Use the General Power Rule, Exponential Rule, or the Chain Rule to compute the derivative.

$$y = \left( 4 - 2 x - 3 x ^ { 2 } \right) ^ { 5 }$$

Solve the equation.

$$\frac{2}{3}+y=2\frac{1}{2}$$

Find the inverse of each function. $y=\log _{2} 4 x$

Find the Jacobian of the transformation T from the uv-plane to the xy-plane. $u=\frac{2 x}{x^{2}+y^{2}}, v=\frac{-2 y}{x^{2}+y^{2}}$

Evaluate each expression.

$$7 b , \text { for } b = 5$$

Graph the parabola using a function grapher.

$$y=-\dfrac{1}{6}x^2$$

In an essay, compare the structures, functions, and processes of the national government, your state government, and your local government in the U.S. federal system.

You have gift card worth $90. You want to buy several movies that cost$12 each. Write and solve an inequality that represents the number of movies you can buy and still have at least $30 on the gift card.

A farmer is planting a straight row of crops with 3 feet between each plant. If 23 are planted, what is the distance between the first and last plant? 72 66 69 63 75

Find an equation of the plane that satisfies the given conditions. Contains (2, 3, -5) and is parallel to $x+y-4 z=1$.

A torpedo 8 m below the surface in 20°C seawater cavitates at a speed of 21 m/s when atmospheric pressure is 101 kPa. If Reynolds number and Froude number effects are negligible, at what speed will it cavitate when running at a depth of 20 m? At what depth should it be to avoid cavitation at 30 m/s?

An individual’s income varies with his or her age. The following table shows the median income I of males of different age groups within the United States for 2009. For each age group, let the class midpoint represent the independent variable, x. For the class “65 years and older,” we will assume that the class midpoint is 69.5.

$$\begin{matrix} \text{Age} & \text{Class Midpoint, x} & \text{Median Income, I}\\ \text{15–24 years} & \text{19.5} & \text{10,778}\\ \text{25–34 years} & \text{29.5} & \text{33,415}\\ \text{35–44 years} & \text{39.5} & \text{44,189}\\ \text{45–54 years} & \text{49.5} & \text{45,540}\\ \text{55–64 years} & \text{59.5} & \text{41,757}\\ \text{65 years and older} & \text{69.5} & \text{25,503}\\ \end{matrix}$$

Use a graphing utility to find the quadratic function of best fit that models the relation between age and median income.