Exam 12: Mathematical Problem Set

Determine whether the vectors $\langle 2,-1\rangle$ and $\langle-2,-5\rangle$ are orthogonal.

What are the foci for the graph of $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1 ?$

The graph of the function $x^{2}+y^{2}-24 x+12 y+100=0$ is a circle. What are the coordinates of the center and the length of the radius?

A student randomly picks one class at a college and gives all the students in the class a survey. Is this a random sample of students at the college?

Rewrite the expression $\frac{\sqrt[5]{x^{7}}}{\sqrt[3]{x^{4}}}$ using a single radical.

The table shows average salaries for employees of Mediocre Tools, Inc. Find a linear regression equation for the data and graph the equation with a scatter plot of the data. (Use x=0 for 1975 , and express values to the nearest hundredth.) Use this data to estimate the average salary in 2008 . $\begin{array}{cc}\text { Year } & \text { Average Salary (\$) } \\\hline 1975 & 15,642 \\1980 & 16,580 \\1985 & 17,409 \\1990 & 19,150 \\1995 & 20,491 \\2000 & 21,925\end{array}$

When a car's brakes are applied, the speed of the car is given by v(t)=80-25t (in feet per second). Use the area under the function to determine how many feet the car travels between the time the brakes are applied (t=0) and the time the car stops.

Pat deposits $\$ 150$ at the end of each month into an account that pays $6 \%$ interest compounded monthly. After 8 years of deposits, what is the account balance?

Write the equation of the ellipse graphed below.

In a certain state park, the number of elk present after t years is modeled by the function $P(t)=\frac{1216}{1+75 e^{-0.03 t}}$ (a) What was the initial population of elk? (b) When will the number of elk be 750 ? (c) What is the maximum number of elk possible in the park?

The matrix $\left[\begin{array}{rrr}1 & 1 & -1 \\-3 & 2 & -1 \\3 & -3 & 2\end{array}\right]$ is the inverse of $\left[\begin{array}{lll}1 & 1 & 1 \\3 & 5 & 4 \\3 & 6 & 5\end{array}\right]$ . Solve the following system by the inverse matrix method. x+y+z=4 3 x+5 y+4 z=3 3 x+6 y+5 z=2

Evaluate all six trigonometric functions of the angle $\theta$ for the triangle given below.

Show that the points A(4,2), B(-1,0), C(-3,5) , and D(2,7) are the vertices of a square.

Find the average rate of change of f(x)=\sin x over the interval $[0, \pi / 2]$

Three sides of a fence and a wall form a rectangular enclosure. The total length of a fence used for the three sides is 240 feet. Let x be the length of the two sides perpendicular to the wall as shown. Write the area A of the enclosure as a function of the length x . Then find the value(s) of x for which the area is 5500 ft² .

Determine whether the vectors $\langle 2,-1\rangle$ and $\langle-3,-7\rangle$ are orthogonal.

Find the partial fraction decomposition of $\frac{2 x+1}{x^{2}} .$

At the top of a radio signal tower there is an antenna with a light on the end of it. From a point on the ground 500 feet from the base of the tower, the angle of elevation to the tip of the light is $35.6^{\circ}$ and the angle of elevation to the bottom of the antenna is $3.4^{\circ}$ What is the height of the antenna, including the light?

What is the equation, in slope-intercept form, of the line that contains the points (-2,7) and (3,5) ?

Which of the following could represent a completegraph of f(x)=ax+x³ whereais a real number?