55
A doctor who is treating a heart patient wants to prescribe medication to lower the patient's blood pressure. The body's reaction to this medication is a function of the dose administered. If the patient takes \(x\) milliliters of the medication, his blood pressure should decrease by \(R = f (x)\) points, where
\begin{equation*}
f (x) = 3x^2  \dfrac{1}{3}x^3
\end{equation*}
For what values of \(x\) is \(R = 0\text{?}\)
Find a suitable domain for the function and explain why you chose this domain.
Graph the function \(f\) on its domain.
How much should the patient's blood pressure drop if he takes \(2\) milliliters of medication?
What is the maximum drop in blood pressure that can be achieved with this medication?
There may be risks associated with a large change in blood pressure. How many milliliters of the medication should be administered to produce half the maximum possible drop in blood pressure?
56
A soup bowl has the shape of a hemisphere of radius \(6\) centimeters. The volume of the soup in the bowl, \(V = f (x)\text{,}\) is a function of the depth, \(x\text{,}\) of the soup.
What is the domain of \(f\text{?}\) Why?

The function \(f\) is given by
\begin{equation*}
f (x) = 6 \pi x^2 \frac{\pi}{3}x^3
\end{equation*}
Graph the function on its domain.
What is the volume of the soup if it is \(3\) centimeters deep?
What is the maximum volume of soup that the bowl can hold?
Find the depth of the soup (to within \(2\) decimal places of accuracy) when the bowl is filled to half its capacity.
57
The population, \(P(t) \text{,}\) of Cyberville has been growing according to the formula
\begin{equation*}
P(t) = t^3  63t^2 + 1403t + 900
\end{equation*}
where \(t\) is the number of years since 1970.

Graph \(P(t)\) in the window
\begin{align*}
{\text{Xmin}} \amp = 0 \amp\amp {\text{Xmax}} = 47\\
{\text{Ymin}} \amp = 0 \amp\amp {\text{Ymax}} = 20,000
\end{align*}
What was the population in 1970? In 1985? In 2004?
By how much did the population grow from 1970 to 1971? From 1985 to 1986? From 2004 to 2005?
Approximately when was the population growing at the slowest rate, that is, when is the graph the least steep?
58
The annual profit, \(P(t)\text{,}\) of the Enviro Company, in thousands of dollars, is given by
\begin{equation*}
P(t) = 2t^3  152t^2 + 3400t + 30
\end{equation*}
where \(t\) is the number of years since 1960, the first year that the company showed a profit.

Graph \(P(t)\) in the window
\begin{align*}
{\text{Xmin}} \amp = 0 \amp\amp {\text{Xmax}} = 94\\
{\text{Ymin}} \amp = 0 \amp\amp {\text{Ymax}} = 50,000
\end{align*}
What was the profit in 1960? In 1980? In 2000?
How did the profit change from 1960 to 1961? From 1980 to 1981? From 2000 to 2001?
During which years did the profit decrease from one year to the next?
59
The total annual cost of educating postgraduate research students at an Australian university, in thousands of dollars, is given by the function
\begin{equation*}
C(x) = 0.0173x^3  0.647x^2 + 9.587x + 195.366
\end{equation*}
where \(x\) is the number of students, in hundreds. (Source: Creedy, Johnson, and Valenzuela, 2002)
Graph the function in a suitable window for up to \(3500\) students.
Describe the concavity of the graph. For what value of \(x\) is the cost growing at the slowest rate?
Approximately how many students can be educated for $\(350,000\text{?}\)
60
It has been proposed that certain cubic functions model the response of wheat and barley to nitrogen fertilizer. These functions exhibit a "plateau" that fits observations better than the standard quadratic model. (See Problem 36 of Section 6.6.) In trials in Denmark, the yield per acre was a function of the amount of nitrogen applied. A typical response function is
\begin{equation*}
Y(x) = 54.45 + 0.305x  0.001655x^2 + 2.935 \times 10^{6}x^3
\end{equation*}
where \(x\) is the amount of fertilizer, in kilograms per acre.(Source: Beattie, Mortensen, and Knudsen, 2005)
Graph the function on the domain \([0, 400]\text{.}\)
Describe the concavity of the graph. In reality, the yield does not increase after reaching its plateau. Give a suitable domain for the model in this application.
Estimate the maximum yield attainable and the optimum application of fertilizer.
61
During an earthquake, Nordhoff Street split in two, and one section shifted up several centimeters. Engineers created a ramp from the lower section to the upper section. In the coordinate system shown in the figure below, the ramp is part of the graph of
\begin{equation*}
y = f (x) = 0.00004x^3  0.006x^2 + 20
\end{equation*}
By how much did the upper section of the street shift during the earthquake?
What is the horizontal distance from the bottom of the ramp to the raised part of the street?
62
The offramp from a highway connects to a parallel oneway road. The accompanying figure shows the highway, the offramp, and the road. The road lies on the \(x\)axis, and the offramp begins at a point on the \(y\)axis. The offramp is part of the graph of the polynomial
\begin{equation*}
y = f (x) = 0.00006x^3  0.009x^2 + 30
\end{equation*}
How far east of the exit does the offramp meet the oneway road?
How far apart are the highway and the road?