Asymptotic Behavior in Real-World Rational Functions

A car rental company charges a flat fee of $50 plus $0.20 per mile driven. Write a rational function to represent the total cost, C, as a function of miles driven, m. What is the asymptotic behavior of this function as m approaches infinity?

• C(m) = 50 + 0.20m; As m approaches infinity, C(m) approaches infinity.

• C(m) = 0.20m; As m approaches infinity, C(m) approaches negative infinity.

• C(m) = 50 + 0.50m; As m approaches infinity, C(m) approaches a constant.

• C(m) = 50 - 0.20m; As m approaches infinity, C(m) approaches zero.

A tank can be filled by two pipes. Pipe A can fill the tank in 4 hours, while Pipe B can fill it in 6 hours. Write a rational function to represent the rate of filling the tank when both pipes are open. Analyze the asymptotic behavior of the tank's filling rate as time increases.

• R(t) = 1/2

• R(t) = 2/3

• R(t) = 3/8

• R(t) = 5/12

A company produces x units of a product, and the cost function is given by C(x) = 100/x + 50. Determine the average cost per unit as x approaches infinity. What does this tell you about the company's cost efficiency?

• The average cost per unit remains constant at 50 as x approaches infinity, suggesting moderate cost efficiency.

• The average cost per unit approaches 100 as x approaches infinity, indicating low cost efficiency.

• The average cost per unit increases without bound as x approaches infinity, indicating poor cost efficiency.

• The average cost per unit approaches 0 as x approaches infinity, indicating high cost efficiency.

A scientist is studying the decay of a radioactive substance. The amount of substance remaining after t days is modeled by the function A(t) = 100/(t + 1). What is the asymptotic behavior of A(t) as t increases?

• A(t) approaches 100 as t increases.

• A(t) approaches 0 as t increases.

• A(t) remains constant at 50 as t increases.

• A(t) approaches infinity as t increases.

A delivery truck travels a distance of d miles at a speed of s miles per hour. If the speed decreases as the distance increases, write a rational function for the time taken, T(d) = d/s(d). How does T(d) behave as d approaches infinity?

• T(d) approaches infinity as d approaches infinity.

• T(d) decreases as d approaches infinity.

• T(d) remains constant as d approaches infinity.

• T(d) approaches zero as d approaches infinity.

A swimming pool is being filled at a rate that decreases over time. The volume of water in the pool after t hours is given by V(t) = 200/(t + 1). What is the asymptotic behavior of V(t) as t increases?

• V(t) increases without bound as t increases.

• V(t) remains constant at 200 as t increases.

• V(t) approaches infinity as t increases.

• V(t) approaches 0 as t increases.

A factory produces widgets at a rate of R(x) = 500/(x + 5), where x is the number of workers. What happens to the production rate as the number of workers increases?

• The production rate fluctuates randomly with the number of workers.

• The production rate remains constant regardless of the number of workers.

• The production rate increases as the number of workers increases.

• The production rate decreases as the number of workers increases.

A student is analyzing the relationship between the number of hours studied and the score achieved on a test. The function S(h) = 100/(h + 1) models this relationship. What is the asymptotic behavior of S(h) as h increases?

• S(h) approaches 0 as h increases.

• S(h) approaches infinity as h increases.

• S(h) remains constant at 50 as h increases.

• S(h) approaches 100 as h increases.

A water tank has a leak that causes the water level to decrease over time. The height of the water level after t hours is modeled by H(t) = 300/(t + 2). What does the asymptotic behavior of H(t) indicate about the water level as time goes on?

• The water level increases indefinitely as time goes on.

• The water level remains constant over time.

• The water level approaches 0 as time goes on.

• The water level oscillates between 100 and 200 units.

A business's profit, P, can be modeled by the function P(x) = 1000/(x + 10), where x is the number of units sold. Analyze the profit as the number of units sold increases. What does this tell you about the business's profitability in the long run?

• The business's profitability increases in the long run as the number of units sold increases.

• The business becomes more profitable with each additional unit sold.

• Profit remains constant regardless of the number of units sold.

• The business's profitability decreases in the long run as the number of units sold increases.