Active Prelude to Calculus

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 \(t\text{.}\)

A different town’s population \(Q\) is given by the function \(Q = g(t) = 42505 - 465t\text{.}\) 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 \(t\text{.}\) 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 \(h = q(t) = 0.65t\text{.}\) 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 $\(10200\text{,}\) and that after \(10\) years its value is $\(4600\text{.}\) Assuming that the car’s value depreciates linearly, find a function \(C = L(t)\) whose output is the value of the car in year \(t\text{.}\) 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.