Chapter 4

Table 4.12 shows monthly life insurance rates, in dollars, for men and women. Let \(m=f(a)\) be the rate for men at age \(a\), and \(w=g(a)\) be the rate for women at age \(a\). (a) Find \(f(65)\). (b) Find \(g(50)\). (c) Solve and interpret \(f(a)=102\). (d) Solve and interpret \(g(a)=57\). (e) For what values of \(a\) is \(f(a)=g(a) ?\) (f) For what values of \(a\) is \(g(a)

Put the functions in the form \(Q=k t\) and state the value of \(k\). $$ Q=\frac{\alpha t-\beta t}{\gamma} $$

One store sells 70 pounds of apples a week, and a second store sells 50 pounds of apples a week. Is the total number of pounds of apples sold, \(a\), proportional to the number of weeks, \(w\) ? If so, what is the constant of proportionality?

Methane is a greenhouse gas implicated as a contributor to global warming. Answer based on the table of values of \(Q=w(t),\) the atmospheric methane level in parts per billion (ppb) \(t\) years after \(1980 .^{5}\) $$\begin{aligned}&\text { Table } 4.15\\\&\begin{array}{c|c|c|c|c|c|c} \hline t & 0 & 5 & 10 & 15 & 20 & 25 \\\\\hline Q & 1575 & 1660 & 1715 & 1750 & 1770 & 1775 \\ \hline\end{array}\end{aligned}$$ Which expression is larger, $$\frac{w(10)-w(0)}{10-0} \quad \text { or } \quad \frac{w(25)-w(10)}{25-10} ?$$ Say what this tells you about atmospheric methane levels.

Table 4.3 shows the 5 top winning teams in the NBA playoffs between 2000 and May 20,2007 and the number of games each team has won. $$ \begin{array}{c|c} \hline \text { Team } & \text { Playoff games won } \\ \hline \text { Lakers } & 66 \\ \hline \text { Spurs } & 66 \\ \hline \text { Pistons } & 61 \\ \hline \text { Nets } & 43 \\ \hline \text { Mavericks } & 41 \\ \hline \end{array} $$ (a) Is the number of games a team won a function of the team? Why or why not? (b) Is the NBA team a function of the number of games won? Why or why not?

Put the functions in the form \(Q=k t\) and state the value of \(k\). $$ Q=(t-3)(t+3)-(t+9)(t-1) $$

The perimeter of a square is proportional to the length of any side. What is the constant of proportionality?

Methane is a greenhouse gas implicated as a contributor to global warming. Answer based on the table of values of \(Q=w(t),\) the atmospheric methane level in parts per billion (ppb) \(t\) years after \(1980 .^{5}\) $$\begin{aligned}&\text { Table } 4.15\\\&\begin{array}{c|c|c|c|c|c|c} \hline t & 0 & 5 & 10 & 15 & 20 & 25 \\\\\hline Q & 1575 & 1660 & 1715 & 1750 & 1770 & 1775 \\ \hline\end{array}\end{aligned}$$ Show that \(\begin{array}{l}\text { Average rate of change } \\ \text { from } 1995 \text { to } 2000\end{array}<\begin{array}{c}\text { Average rate of change } \\ \text { from } 2000 \text { to } 2005 .\end{array}\) Does this mean the average methane level is going down? If not, what does it mean?

Let \(r(p)\) be the revenue in dollars that a company receives when it charges \(p\) dollars for a product. Explain the meaning of the following statements. (a) \(r(15)=112,500\) (b) \(\quad r(a)=0\) (c) \(r(1)=b\) (d) \(c=r(p)\)

The price of apartments near a subway is given by $$ \text { Price }=\frac{1000 \cdot A}{10 d} \text { dollars, } $$ where \(A\) is the area of the apartment in square feet and \(d\) is the distance in miles from the subway. Which letters are constants and which are variables if (a) You want an apartment of 1000 square feet? (b) You want an apartment 1 mile from the subway? (c) You want an apartment that costs \(\$ 200,000 ?\)

A bike shop's revenue is directly proportional to the number of bicycles sold. When 50 bicycles are sold, the revenue is \(\$ 20,000\). (a) What is the constant of proportionality, and what are its units? (b) What is the revenue if 75 are sold?

Methane is a greenhouse gas implicated as a contributor to global warming. Answer based on the table of values of \(Q=w(t),\) the atmospheric methane level in parts per billion (ppb) \(t\) years after \(1980 .^{5}\) $$\begin{aligned}&\text { Table } 4.15\\\&\begin{array}{c|c|c|c|c|c|c} \hline t & 0 & 5 & 10 & 15 & 20 & 25 \\\\\hline Q & 1575 & 1660 & 1715 & 1750 & 1770 & 1775 \\ \hline\end{array}\end{aligned}$$ Project the value of \(w(-20)\) by assuming $$\frac{w(0)-w(-20)}{20}=\frac{w(10)-w(0)}{10}$$ Explain the assumption that goes into making your projection and what your answer tells you about atmospheric methane levels.

Antonio and Lucia are both driving through the desert from Tucson to San Diego, which takes each of them 7 hours of driving time. Antonio's car starts out full with 14 gallons of gas and uses 2 gallons per hour. Lucia's SUV starts out full with 30 gallons of gas and uses 6 gallons per hour. (a) Construct a table showing how much gas is in each of their tanks at the end of each hour into the trip. Assume each stops for gas just as the tank is empty, and then the tank is filled instantaneously. (b) Use your table to determine when they have the same amount of gas. (c) If they drive at the same speed while driving and only stop for gas, which of them gets to San Diego first? (Assume filling up takes time.) (d) Now suppose that between 1 hour and 6.5 hours outside of Tucson, all of the gas stations are closed unexpectedly. Does Antonio arrive in San Diego? Does Lucia? (e) The amount of gas in Antonio's tank after \(t\) hours is \(14-2 t\) gallons, and the amount in Lucia's tank is \(30-6 t\) gallons. When does (i) \(14-2 t=30-6 t ?\) (ii) \(14-2 t=0 ?\) (iii) \(30-6 t=0 ?\)

Let \(d(v)\) be the braking distance in feet of a car traveling at \(v\) miles per hour. Explain the meaning of the following statements. (a) \(d(30)=111\) (b) \(\quad d(a)=10\) (c) \(d(10)=b\) (d) \(s=d(v)\)

The total cost of purchasing gasoline for your car is directly proportional to the the number of gallons pumped, and 11 gallons cost \(\$ 36.63\). (a) What is the constant of proportionality, and what are its units? (b) How much do 15 gallons cost?

The required cooling capacity, in BTUs, for a room air conditioner is proportional to the area of the room being cooled. A room of 280 square feet requires an air conditioner whose cooling capacity is 5600 BTUs. (a) What is the constant of proportionality, and what are its units? (b) If an air conditioner has a cooling capacity of 10,000 BTUs, how large a room can it cool?

You deposit \(\$ P\) into a bank where it earns \(2 \%\) interest per year for 10 years. Use the formula \(B=P(1+r)^{t}\) for the balance \(\$ B,\) where \(r\) is the interest rate (written as a decimal) and \(t\) is time in years. (a) Explain why \(B\) is proportional to \(P\). What is the constant of proportionality? (b) Is \(P\) proportional to \(B ?\) If so, what is the constant of proportionality?

The length \(m\), in inches, of a model train is proportional to the length \(r,\) in inches, of the corresponding real train. (a) Write a formula expressing \(m\) as a function of \(r\). (b) An HO train is \(1 / 87^{\text {th }}\) the size of a real train. What is the constant of proportionality? What is the length in feet of a real locomotive if the \(\mathrm{HO}\) locomotive is 10.5 inches long? (c) A Z scale train is \(1 / 220^{\text {th }}\) the size of a real train. What is the constant of proportionality? What is the length, in inches, of a \(Z\) scale locomotive if the real locomotive is 75 feet long?

The cost of denim fabric is directly proportional to the amount that you buy. Let \(C\) be the cost, in dollars, of \(x\) yards of denim fabric. (a) Write a formula expressing \(C\) as a function of \(x\). (b) One type of denim costs \(\$ 28.50\) for 3 yards. Find the constant of proportionality and give its units. (c) How much does 5.5 yards of this type of denim cost?

The distance \(M,\) in inches, between two points on a map is proportional to the actual distance \(d\), in miles, between the two corresponding locations. (a) If \(1 / 2\) inch represents 5 miles, find the constant of proportionality and give its units.

The blood mass of a mammal is proportional to its body mass. A rhinoceros with body mass 3000 kilograms has blood mass of 150 kilograms. Find a formula for the blood mass of a mammal in terms of the body mass and estimate the blood mass of a human with body mass 70 kilograms.

The distance a car travels on the highway is proportional to the quantity of gas consumed. A car travels 225 miles on 5 gallons of gas. Find the constant of proportionality, give units for it, and explain its meaning.

Evaluate the expressions in Problems 41-42 given that $$ f(n)=\frac{1}{2} n(n+1) $$ $$ f(n+1)-f(n) $$