1.4 Linear Functions (continued)

Interpreting linear functions in context

Summary

The Dolbear function $T = D(N) = 40 + 0.25N$ from Section 1.2 is a linear function whose slope is $m = 0.25$. What is the meaning of the slope in this context?

Suppose that the amount of ice cover at the peak of Mt. Kilimanjaro is changing at a constant average rate from year to year. Find a linear model $A = f(t)$ whose output is the area of the ice cover, $A$, in square meters in year $t$ (where $t$ is the number of years after 2000).

What do the slope and $A$-intercept mean in the model you found in (a)? In particular, what are the units on the slope?

Compute $f(17)$. What does this quantity measure? Write a complete sentence to explain.

If the model holds further into the future, when do we predict the ice cover will vanish?

In light of your work above, what is a reasonable domain to use for the model $A = f(t)$? What is the corresponding range?

A town's population initially has $28750$ people present and then grows at a constant rate of $825$ people per year. Find a linear model $P = f(t)$ for the number of people in the town in year $t$.

A different town's population $Q$ is given by the function $Q = g(t) = 42505 - 465t$. What is the slope of this function and what is its meaning in the model? Write a complete sentence to explain.

A spherical tank is being drained with a pump. Initially the tank is full with $\frac{32\pi}{3}$ cubic feet of water. Assume the tank is drained at a constant rate of $1.2$ cubic feet per minute. Find a linear model $V = p(t)$ for the total amount of water in the tank at time $t$. In addition, what is a reasonable approximate domain for the model?

A conical tank is being filled in such a way that the height of the water in the tank, $h$ (in feet), at time $t$ (in minutes) is given by the function $h = q(t) = 0.65t$. What can you say about how the water level is rising? Write at least one careful sentence to explain.

Suppose we know that a $5$-year old car's value is ＄$10200$, and that after $10$ years its value is ＄$4600$. Assuming that the car's value depreciates linearly, find a function $C = L(t)$ whose output is the value of the car in year $t$. What is a reasonable domain for the model? What is the value and meaning of the slope of the line? Write at least one careful sentence to explain.

An apartment manager keeps careful record of how the rent charged per unit corresponds to the number of occupied units in a large complex. The collected data is shown in Table.（This problem is a slightly modified version of one found in http://mathquest.carroll.edu/CarrollActiveCalculus/C_0.html.）

Monthly Rent | ＄650 | ＄700 | ＄750 | ＄800 | ＄850 | ＄900

Occupied Apartments | 203 | 196 | 189 | 182 | 175 | 168

• Why is it reasonable to say that the number of occupied apartments is a linear function of rent? - Let $A$ be the number of occupied apartments and $R$ the monthly rent charged (in dollars). If we let $A = f(R)$, what is the slope of the linear function $f$? What is the meaning of the slope in the context of this question? - Determine a formula for $A = f(R)$. What do you think is a reasonable domain for the function? Why? - If the rent were to be increased to ＄1000, how many occupied apartments should the apartment manager expect? How much total revenue would the manager collect in a given month when rent is set at ＄1000? - Why do you think the apartment manager is interested in the data that has been collected?

Alicia and Dexter are each walking on a straight path. For a particular $10$-second window of time, each has their velocity (in feet per second) measured and recorded as a function of time. Their respective velocity functions are plotted in Figure.

• Determine formulas for both $A = f(t)$ and $D = g(t)$. - What is the value and meaning of the slope of $A$? Write a complete sentence to explain and be sure to include units in your response. - What is the value and meaning of the average rate of change of $D$ on the interval $[4,8]$? Write a complete sentence to explain and be sure to include units in your response. - Is there ever a time when Alicia and Damon are walking at the same velocity? If yes, determine both the time and velocity; if not, explain why. - Is is possible to determine if there is ever a time when Alicia and Damon are located at the same place on the path? If yes, determine the time and location; if not, explain why not enough information is provided.

An inverted conical tank with depth $4$ feet and radius $2$ feet is completely full of water. The tank is being drained by a pump in such a way that the amount of water in the tank is decreasing at a constant rate of $1.5$ cubic feet per minute. Let $V = f(t)$ denote the volume of water in the tank at time $t$ and $h = g(t)$ the depth of the water in the tank at time $t$, where $t$ is measured in minutes.

• How much water is in the tank at $t = 0$ when the tank is completely full? - Explain why volume, $V$, when viewed as a function of time, $t$, is a linear function. - Determine a formula for $V = f(t)$. - At what exact time will the tank be empty? - What is a reasonable domain to use for the model $f$? What is its corresponding range?