Sheryl E.

From 1950 through 2005, the per capita consumption $ C $ of cigarettes by Americans (age 18 and older) can be modeled by $ C = 3565.0 + 60.30t - 1.783t^2, 0 \le t \le 55 $, where $ t $ is the year, with $ t = 0 $ corresponding to 1950.

(a) Use a graphing utility to graph the model.
(b) Use the graph of the model to approximate the maximum average annual consumption. Beginning in 1966, all cigarette packages were required by law to carry a health warning. Do you think the warning
had any effect? Explain.
(c) In 2005, the U.S. population (age 18 and over) was 296,329,000. Of those, about 59,858,458 were
smokers. What was the average annual cigarette consumption per smoker in 2005? What was the
average daily cigarette consumption per smoker?

An open box with locking tabs is to be made from a square piece of material 24 inches on a side. This is to be done by cutting equal squares from the corners and folding along the dashed lines shown in the figure.

(a) Write a function $ V(x) $ that represents the volume of the box.
(b) Determine the domain of the function $ V $.
(c) Sketch a graph of the function and estimate the value of for which $ V(x) $ is maximum.

The table shows the average daily high temperatures in Houston $ H $ (in degrees Fahrenheit) for month $ t $, with $ t = 1 $ corresponding to January. (Source: National Climatic Data Center)

(a) Create a scatter plot of the data.

(b) Find a cosine model for the temperatures in Houston.

(c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data?

(d) What is the overall average daily high temperature in Houston?

(e) Use a graphing utility to describe the months during which the average daily high temperature is above $ 86^\circ $ and below $ 86^\circ F $.

TRUSSES $Q$ is the midpoint of the line segment $\overline{PR}$ in the truss rafter shown in the figure. What are the lengths of the line segments $\overline{PQ}$, $\overline{QS}$, and $\overline{RS}$?

Fill in the blank to complete the trigonometric formula.

$ \cos u - \cos v $ = ________

When two railroad tracks merge, the overlapping portions of the tracks are in the shapes of circular arcs (see figure). The radius of each arc $ r $ (in feet) and the angle $ \theta $ are related by

$ \dfrac{x}{2} = 2r \sin^2 \dfrac{\theta}{2} $

Write a formula for $ x $ in terms of $ \cos \theta $.