IAR - Illinois Assessment of Readiness Math Modeling and Application Questions and Answers

Question 1

A biologist is modeling the population of a certain species of fish in a lake.
The initial population is 1,500.
The population is expected to grow by 4.5% each year.
The biologist wants to create a function, P(t), to model the fish population after t years.
Which of the following functions best represents this scenario?

Accepted Answer

P(t) = 1500(1.045)^t

Answer

P(t) = 1500(0.045)^t

Answer

P(t) = 1500 + 1.045t

Answer

P(t) = 1500 + 0.045t

Question 2

A city is planning a new rectangular park. The park's length must be 30 feet longer than its width. The city's ordinance requires that the park's area be at least 5,400 square feet. If 'w' represents the width of the park in feet, which inequality correctly models the constraint on the park's dimensions?

Accepted Answer

w(w + 30) ≥ 5400

Answer

w + (w + 30) ≥ 5400

Answer

2w + 2(w + 30) ≥ 5400

Answer

w(w - 30) ≤ 5400

Question 3

The function H(t) = -16t^2 + 80t + 4 models the height, H, in feet, of a ball t seconds after it is thrown from a platform. What does the constant term '4' represent in the context of this model?

Accepted Answer

The initial height of the ball.

Answer

The maximum height the ball reaches.

Answer

The time it takes for the ball to hit the ground.

Answer

The initial velocity of the ball.

Question 4

A small business is selling handmade scarves. The owner determines that the profit, P(x), from selling x scarves can be modeled by the function P(x) = -0.5x^2 + 60x - 200. When interpreting this model, which of the following is an appropriate step in the modeling process?

Accepted Answer

Finding the vertex of the parabola to determine the number of scarves that must be sold to maximize profit.

Answer

Assuming the model is valid for any number of scarves, including negative values.

Answer

Concluding that the business will make a profit as long as they sell at least one scarf.

Answer

Using the model to predict profit for selling 1,000,000 scarves without considering production capacity.

Question 5

A car rental company charges a flat fee of $45 plus $0.25 per mile driven. A customer has a budget of $100 for the rental. To model this situation, a student writes the inequality 45 + 0.25m ≤ 100, where 'm' is the number of miles driven. What is the most reasonable and practical constraint on the variable 'm' in this real-world application?

Accepted Answer

0 ≤ m ≤ 220

Answer

m can be any real number.

Answer

m ≤ 220

Answer

m ≥ 0

Question 6

To track the spread of a new mobile app, its developers use the logistic growth model N(t) = 250,000 / (1 + 500e^(-0.8t)), where N(t) is the number of users after t days. Which of the following statements best describes the meaning of the value 250,000 in the model?

Accepted Answer

The approximate maximum number of users the app can expect to have.

Answer

The initial number of app users on day zero.

Answer

The rate at which the number of users is growing each day.

Answer

The number of days it takes to reach the first 250,000 users.