A company tracks its revenue over several years and finds that it can be modeled by a polynomial function. The revenue $R(t)$ in millions of dollars, where $t$ is the number of years since the company started, is given by $R(t)=-2t^3+15t^2+10t+5$. Determine the year in which the company will achieve its maximum revenue.

A biologist is studying the population growth of a certain species of fish in a lake. The population $P(t)$ after $t$ years can be modeled by the rational function $P(t)=\frac{500t}{t+5}$. Determine the horizontal asymptote of this function and explain its significance in the context of the fish population.

A car's value depreciates over time and can be modeled by the polynomial function $V(t)=-2000t^2+30000t+20000$, where $V(t)$ is the value in dollars and $t$ is the number of years since purchase. Determine the time at which the car's value is at its maximum and the maximum value.

A scientist is modeling the concentration of a drug in the bloodstream over time using the rational function $C(t)=\frac{50t}{t^2+4}$, where $C(t)$ is the concentration in mg/L and $t$ is the time in hours. Determine the time at which the concentration is at its maximum and the maximum concentration.

A farmer is modeling the yield of a crop based on the amount of fertilizer used. The yield $Y(x)$ in bushels can be modeled by the polynomial function $Y(x)=-0.5x^3+4x^2+20x+100$, where $x$ is the amount of fertilizer in pounds. Determine the amount of fertilizer that maximizes the yield and the maximum yield.

A polynomial function $P(x)=2x^3-3x^2+x-5$ is constructed to model the growth of a plant over time. Given the actual data points $(1,-5),(2,-3),(3,10)$, evaluate the accuracy of the model by comparing the polynomial's predictions to the actual data.

A rational function $R(x)=\frac{2x^2-3x+1}{x-2}$ is proposed to model the speed of a car over time. Given the actual speed data points $(1,0),(3,4),(4,5)$, evaluate the appropriateness of the model by comparing the function's predictions to the actual data.

A polynomial function $Q(x)=x^4-2x^3+x^2-x+1$ is used to model the temperature changes over a day. Given the actual temperature data points $(0,1),(1,0),(2,5)$, evaluate the accuracy of the model by comparing the polynomial's predictions to the actual data.

A rational function $S(x)=\frac{x^2-4}{x+2}$ is proposed to model the population growth of a species. Given the actual population data points $(1,-1.5),(3,1),(5,3)$, evaluate the appropriateness of the model by comparing the function's predictions to the actual data.

A polynomial function $T(x)=3x^3-2x^2+x-4$ is used to model the revenue of a company over time. Given the actual revenue data points $(1,-2),(2,6),(3,20)$, evaluate the accuracy of the model by comparing the polynomial's predictions to the actual data.

A company manufactures and sells widgets. The cost to produce $x$ widgets is given by the function $C(x)=5x+200$, and the revenue generated from selling $x$ widgets is given by the function $R(x)=15x$. Determine the break-even point, where the cost equals the revenue.

A farmer is planning to fence a rectangular field along a river. The river acts as one side of the rectangle, so the farmer only needs to fence the other three sides. If the farmer has 300 meters of fencing available, express the area $A$ of the field as a function of the width $w$ of the field.

A car rental company charges a flat fee of $50 plus $0.20 per mile driven. Write a function $C(x)$ that represents the total cost of renting a car and driving $x$ miles. Then, calculate the cost for driving 150 miles.

A company's profit $P(x)$ in thousands of dollars is modeled by the function $P(x)=-2x^2+12x-16$, where $x$ is the number of units sold in thousands. Determine the number of units sold that maximizes the profit.