Exam 4: Polynomial and Rational Functions

An arch has the shape of the parabola A rectangle is fit under the arch by selecting a point on the parabola (see the figure). If the rectangle has base and height . Find the base of a second rectangle that has the same area.

Use the factor theorem to decide whether is a factor of the polynomial. ;

Sketch the graph of .

Applying the first theorem on bounds for real zeros of polynomials, determine the smallest and largest integers that are upper and lower bounds, respectively, for the real solutions of the equation. With the aid of a graphing utility, discuss the validity of the bounds.

Find the fourth-degree polynomial function whose graph is shown in the figure.

Find all values of such that

For the following function, find a number such that the graph of contains the point

A flat metal plate is positioned in an xy-plane such that the temperature T (in C) at the point (x, y) is inversely proportional to the distance from the origin. If the temperature at the point P (3, 4) is 20 C, find the temperature at the point Q (6, 5).

Find the polynomial function of degree whose graph is shown in the figure.

A polynomial f ( x ) with real coefficients and leading coefficient 1 has the given zeros and degree. Express f ( x ) as a product of linear and quadratic polynomials with real coefficients that are irreducible over R.

Use synthetic division to decide whether is a zero of . ;

A meteorologist determines that the temperature T for a certain 24-hour period in winter was given by the following formula. is time in hours and corresponds to 6 A.M.

The density D(h) (in kg/m ³ ) of the earth's atmosphere at an altitude of h meters can be approximated by where and . Use a graphing utility to graph D and approximate the altitude h at which the density is 0.6.

Find all solutions of the equation.

The density D(h) (in kg/m ³ ) of the earth's atmosphere at an altitude of h meters can be approximated by where and . Use a graphing utility to graph D and approximate the altitude h at which the density is 0.3.

When uranium disintegrates into lead, one step in the process is the radioactive decay of radium into radon gas. Radon enters through the soil into home basements, where it presents a health hazard if inhaled. In the simplest case of radon detection, a sample of air with volume V is taken. After equilibrium has been established, the radioactive decay D of the radon gas is counted with efficiency E over time t. The radon concentration C present in the sample of air varies directly as the product of D and E and inversely as the product of V and t. For a fixed radon concentration C and time t, find the change in the radioactive decay count D if V is multiplied by 4 and E is reduced by 20%.

The density D(h) (in kg/m ³ ) of the earth's atmosphere at an altitude of h meters can be approximated by where and . Use a graphing utility to graph D and approximate the altitude h at which the density is 0.1.

Express the statement as a formula that involves the variables w, z, u and a constant of proportionality k, and then determine the value of k from the condition : w varies directly as z and inversely as the square root of u, if z = 2 and u = 36, then w = 12

Salt water of concentration 0.1 pound of salt per gallon flows into a large tank that initially contains 40 gallons of pure water. If the flow rate of salt water into the tank is 5 gal/min, find a formula for the salt concentration (in lb/gal) after t minutes.

Sketch the graph of .