Math Modeling and Application Flashcards

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?

Answer: P(t) = 1500(1.045)^t

This scenario describes exponential growth. The initial value is 1500. The growth rate is 4.5%, or 0.045. The growth factor is 1 + the growth rate, which is 1 + 0.045 = 1.045. The general form for an exponential growth function is y = a(1+r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time. Therefore, the correct function is P(t) = 1500(1.045)^t.

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?

Answer: w(w + 30) ≥ 5400

The area of a rectangle is calculated by multiplying its length and width. The width is 'w'. The length is given as '30 feet longer than its width', which can be expressed as 'w + 30'. The area must be 'at least 5,400 square feet', which means the area is greater than or equal to 5400. Therefore, the inequality that models this situation is w(w + 30) ≥ 5400.

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?

Answer: The initial height of the ball.

In a quadratic model of projectile motion like this one, the constant term represents the initial value of the function, which occurs when t=0. In this context, H(0) is the height of the ball at time t=0, which is the moment it is thrown. Therefore, the '4' represents the initial height of the ball, which is 4 feet.

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?

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

The modeling process involves analyzing the mathematical model to draw conclusions about the real-world situation. Since this function is a downward-opening parabola, its vertex represents the maximum point. In this context, the x-coordinate of the vertex would represent the number of scarves needed to achieve the maximum profit, and the y-coordinate would be that maximum profit. The other options represent misinterpretations or invalid applications of the model; for instance, selling negative scarves is impossible, and the model has constraints on its valid domain.

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?

Answer: 0 ≤ m ≤ 220

In the context of the problem, 'm' represents the number of miles driven, which cannot be a negative number, so m ≥ 0. Solving the inequality 45 + 0.25m ≤ 100 gives 0.25m ≤ 55, which simplifies to m ≤ 220. Combining the practical constraint (miles cannot be negative) and the budgetary constraint (miles are limited by the total cost), the most reasonable and complete constraint for 'm' is 0 ≤ m ≤ 220.

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?

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

The given function is a logistic model, which is often used to represent growth that has a limiting factor. The general form is N(t) = K / (1 + Ae^(-kt)), where 'K' is the carrying capacity, or the maximum sustainable value. As t becomes very large, the e^(-0.8t) term approaches 0, and the function N(t) approaches 250,000 / 1. Therefore, 250,000 represents the carrying capacity, or the approximate maximum number of users.