91Ó°ÊÓ

An object falling vertically through the air is subjected to viscous resistance as well as to the force of gravity. Assume that an object with mass \(m\) is dropped from a height \(s_{0}\) and that the height of the object after \(t\) seconds is $$ s(t)=s_{0}-\frac{m g}{k} t+\frac{m^{2} g}{k^{2}}\left(1-e^{-k t / m}\right) $$ where \(g=32.17 \mathrm{ft} / \mathrm{s}^{2}\) and \(k\) represents the coefficient of air resistance in \(\mathrm{lb}-\mathrm{s} / \mathrm{ft}\). Suppose \(s_{0}=300 \mathrm{ft}\), \(m=0.25 \mathrm{lb}\), and \(k=0.1 \mathrm{lb}-\mathrm{s} / \mathrm{ft}\). Find, to within \(0.01 \mathrm{~s}\), the time it takes this quarter-pounder to hit the ground.

Let \(g \in C^{1}[a, b]\) and \(p\) be in \((a, b)\) with \(g(p)=p\) and \(\left|g^{\prime}(p)\right|>1\). Show that there exists a \(\delta>0\) such that if \(0<\left|p_{0}-p\right|<\delta\), then \(\left|p_{0}-p\right|<\left|p_{1}-p\right|\). Thus, no matter how close the initial approximation \(p_{0}\) is to \(p\), the next iterate \(p_{1}\) is farther away, so the fixed-point iteration does not converge if \(p_{0} \neq p\).

The accumulated value of a savings account based on regular periodic payments can be determined from the annuity due equation, $$ A=\frac{P}{i}\left[(1+i)^{n}-1\right] $$ In this equation, \(A\) is the amount in the account, \(P\) is the amount regularly deposited, and \(i\) is the rate of interest per period for the \(n\) deposit periods. An engineer would like to have a savings account valued at \(\$ 750,000\) upon retirement in 20 years and can afford to put \(\$ 1500\) per month toward this goal. What is the minimal interest rate at which this amount can be invested, assuming that the interest is compounded monthly?

A drug administered to a patient produces a concentration in the blood stream given by \(c(t)=A t e^{-t / 3}\) milligrams per milliliter, \(t\) hours after \(A\) units have been injected. The maximum safe concentration is \(1 \mathrm{mg} / \mathrm{mL}\). a. What amount should be injected to reach this maximum safe concentration, and when does this maximum occur? b. An additional amount of this drug is to be administered to the patient after the concentration falls to \(0.25 \mathrm{mg} / \mathrm{mL}\). Determine, to the nearest minute, when this second injection should be given. c. Assume that the concentration from consecutive injections is additive and that \(75 \%\) of the amount originally injected is administered in the second injection. When is it time for the third injection?

Player A will shut out (win by a score of \(21-0\) ) player B in a game of racquetball with probability $$ P=\frac{1+p}{2}\left(\frac{p}{1-p+p^{2}}\right)^{21} $$ where \(p\) denotes the probability A will win any specific rally (independent of the server). (See [Keller, J], p. 267.) Determine, to within \(10^{-3}\), the minimal value of \(p\) that will ensure that A will shut out B in at least half the matches they play.

In the design of all-terrain vehicles, it is necessary to consider the failure of the vehicle when attempting to negotiate two types of obstacles. One type of failure is called hang-up failure and occurs when the vehicle attempts to cross an obstacle that causes the bottom of the vehicle to touch the ground. The other type of failure is called nose-in failure and occurs when the vehicle descends into a ditch and its nose touches the ground. The accompanying figure, adapted from [Bek], shows the components associated with the nosein failure of a vehicle. In that reference it is shown that the maximum angle \(\alpha\) that can be negotiated by a vehicle when \(\beta\) is the maximum angle at which hang-up failure does not occur satisfies the equation $$ A \sin \alpha \cos \alpha+B \sin ^{2} \alpha-C \cos \alpha-E \sin \alpha=0 $$ where $$ \begin{aligned} &A=l \sin \beta_{1}, \quad B=l \cos \beta_{1}, \quad C=(h+0.5 D) \sin \beta_{1}-0.5 D \tan \beta_{1} \\ &\text { and } E=(h+0.5 D) \cos \beta_{1}-0.5 D \end{aligned} $$ a. It is stated that when \(l=89\) in., \(h=49\) in., \(D=55\) in., and \(\beta_{1}=11.5^{\circ}\), angle \(\alpha\) is approximately \(33^{\circ}\). Verify this result. b. Find \(\alpha\) for the situation when \(l, h\), and \(\beta_{1}\) are the same as in part (a) but \(D=30\) in.