Charles C.

In Example 5, we modeled a measles pathogenesis curve by a function $ f $. A patient infected with the measles virus who has some immunity to the virus has a pathogenesis curve that can be modeled by, for instance, $ g(t) = 0.9 f(t) $.
(a) If the same threshold concentration of the virus is required for infectiousness to begin as in Example 5, on what day does this occur?
(b) Let $ P_3 $ be the point of the graph of $ g $ where infectiousness begin. It has been shown that infectiousness ends at a point $ P_4 $ on the graph of $ g $ where the line through $ P_3 $, $ P_4 $ has the same slope as the line through $ P_1 $, $ P_2 $ in Example 5(b). On what day does infectiousness end?
(c) Compute the level of infectiousness for this patient.

Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by side.

A model for tumor growth is given by the Gompertz equation
$ \frac {dV}{dt} = a (\ln b - \ln V) V $
where $ a $ and $ b $ are positive constant and $ V $ is the volume of the tumor measured in $ mm^3. $
(a) Find a family of solution for tumor volume as a function of time.
(b) Find the solution that has an initial tumor volume of $ V(0) = 1 mm^3. $

In Section 9.3 we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable differentiable equations. (See Example 6 in that section.) If the rates of flow into and out of the system are different, then the volume is not constant and the resulting differential equation is linear but not separable.
A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at a rate of 3 L/min. If $ y(t) $ is the amount of salt (in kilograms) after $ t $ minutes, show that $ y $ satisfies the differential equation
$ \frac {dy}{dt} = 2 - \frac {3y}{100 + 2t} $
Solve this equation and find the concentration after 20 minutes.

A roofing contractor is fabricating gutters from 12-inch aluminum sheeting. The contractor plans to use an aluminum siding folding press to create the gutter by creasing equal lengths for the sidewalls (see figure).

(a) Let $ x $ represent the height of the sidewall of the gutter. Write a function $ A $ that represents the cross-sectional area of the gutter.
(b) The length of the aluminum sheeting is 16 feet.Write a function $ V $ that represents the volume of one run of gutter in terms of $ x $.
(c) Determine the domain of the function in part (b).
(d) Use a graphing utility to create a table that shows sidewall heights $ x $ and the corresponding volumes $ V $.Use the table to estimate the dimensions that will produce a maximum volume.
(e) Use a graphing utility to graph $ V $. Use the graph to estimate the value of $ x $ for which $ V(x) $ is a maximum. Compare your result with that of part (d).
(f) Would the value of $ x $ change if the aluminum sheeting were of different lengths? Explain.

In Exercises 7 - 10, determine whether each ordered pair is a solution of the system of equations

$ \left\{\begin{array}{l} 2x - y = 4\\8x + y = -9\end{array}\right. $

(a) $ (0 , -4) $
(b) $ (-2 , 7) $
(c) $ (\dfrac{3}{2} , -1) $
(d) $ (- \dfrac{1}{2} , -5) $