Interpreting Nonlinear Functions Flashcards

The height `h`, in meters, of a ball thrown upwards from a building is modeled by the function `h(t) = -4.9t^2 + 20t + 40`, where `t` is the time in seconds after the ball is thrown. Based on the model, what is the meaning of the number 40 in this context?

Answer: The initial height from which the ball was thrown.

In this quadratic function, the variable `t` represents time. When `t=0`, the ball has just been thrown. Plugging `t=0` into the equation gives `h(0) = -4.9(0)^2 + 20(0) + 40 = 40`. Therefore, 40 represents the initial height of the ball, in meters, when it was thrown.

The number of bacteria in a colony is modeled by the function `P(t) = 500(1.08)^t`, where `t` is the time in hours. Which of the following best describes the meaning of the number 1.08 in the function?

Answer: The number of bacteria increases by 8% each hour.

This is an exponential growth model of the form `A(1+r)^t`. The base of the exponent, `(1+r)`, is the growth factor. Here, the growth factor is 1.08, which means the population is multiplied by 1.08 each hour. This corresponds to a growth rate of 0.08, or 8%, per hour.

The profit, `P`, in thousands of dollars, from selling a product at a price of `x` dollars per unit is given by the function `P(x) = -2x^2 + 80x - 600`. The graph of this function is a parabola that opens downward. What does the vertex of the parabola represent?

Answer: The price that results in the maximum profit.

The graph of the profit function is a downward-opening parabola. The vertex of such a parabola represents the maximum point on the graph. In this context, the x-coordinate of the vertex is the price that yields the highest profit, and the y-coordinate is the maximum possible profit.

The value of a car, `V`, in dollars, `t` years after its purchase is modeled by the function `V(t) = 25000(0.85)^t`. A graph of this function is created. What does the y-intercept of the graph represent in this context?

Answer: The initial purchase price of the car.

The y-intercept of a graph is the point where the x-value (in this case, `t`) is 0. For this function, `V(0) = 25000(0.85)^0 = 25000(1) = 25000`. This means that at time `t=0`, which is the moment of purchase, the car's value was $25,000. Therefore, the y-intercept represents the initial purchase price of the car.

The amount of a certain medication, `A`, in milligrams, in a patient's bloodstream `t` hours after administration is given by `A(t) = 100(0.5)^(t/4)`. What does the number 4 represent in this function?

Answer: The half-life of the medication in hours.

This is an exponential decay function representing half-life. The general form is `A(t) = A_0 (1/2)^(t/H)`, where `H` is the half-life. In this function, the amount of medication is halved every time `t` increases by 4, meaning it takes 4 hours for half of the medication to be eliminated from the bloodstream.

The height of a rocket, `h`, in feet, `t` seconds after launch is modeled by `h(t) = -16t^2 + 224t`. At what time `t` does the rocket reach its maximum height?

Answer: 7 seconds

The function is a quadratic `at^2 + bt + c`, and its graph is a parabola. The maximum height occurs at the vertex. The t-coordinate of the vertex of a parabola is given by the formula `t = -b / (2a)`. In this equation, a = -16 and b = 224, so `t = -224 / (2 * -16) = -224 / -32 = 7`.

A researcher models the spread of a rumor in a school of 1000 students using the function `N(t) = 1000 / (1 + 999e^(-0.5t))`, where `N(t)` is the number of students who have heard the rumor after `t` days. According to the model, what is the best interpretation of the number 1000?

Answer: The maximum number of students who can hear the rumor.

This is a logistic growth model, which describes a population that grows rapidly at first and then levels off. The value in the numerator, often called the carrying capacity, represents the maximum possible value of the function. In this context, it represents the total population, which is the maximum number of students who can possibly hear the rumor.