Free SAT Interpreting Nonlinear Functions Questions and Answers 1 — Questions and Answers
Question 1: 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?
- The maximum height the ball reaches.
- The initial height from which the ball was thrown. (Correct answer)
- The time it takes for the ball to hit the ground.
- The initial speed of the ball.
Correct 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.
Question 2: 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?
- The initial number of bacteria was 108.
- The number of bacteria increases by 1.08 each hour.
- The number of bacteria increases by 8% each hour. (Correct answer)
- The number of bacteria reaches its maximum in 1.08 hours.
Correct 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.
Question 3: 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?
- The price at which the profit is zero.
- The initial cost of producing the product.
- The price that results in the maximum profit. (Correct answer)
- The number of units sold.
Correct 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.
Question 4: 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?
- The value of the car after one year.
- The annual rate of depreciation of the car.
- The number of years it takes for the car's value to be zero.
- The initial purchase price of the car. (Correct answer)
Correct 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.
Question 5: 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?
- The initial dose of the medication in milligrams.
- The number of hours it takes for the medication to be completely eliminated.
- The half-life of the medication in hours. (Correct answer)
- The percentage of medication that decays each hour.
Correct 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.
Question 6: 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?
- 4 seconds
- 7 seconds (Correct answer)
- 14 seconds
- 224 seconds
Correct 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`.
Question 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?
- The initial number of students who started the rumor.
- The number of days it takes for the rumor to spread to half the students.
- The maximum number of students who can hear the rumor. (Correct answer)
- The rate at which the rumor is spreading.
Correct 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.
Free SAT Interpreting Nonlinear Functions Questions and Answers 1
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?