Question

company made a change in its advertising. suppose the companys monthly revenue (in thousands of dollars) x months after the change can be modeled with an exponential function. the graph of this function is shown below. use the model to answer the parts to the right. (a) what is the initial monthly revenue of the company? thousand dollars the initial revenue is represented by the select of the graph. (b) for the first 10 months, as time increases, the monthly revenue select. (c) give the equation of the asymptote. y = choose the statement that best describes the meaning of the asymptote. the companys monthly revenue does not rise above 100 thousand dollars per month. the companys monthly revenue does not fall below 20 thousand dollars per month. the company will only stay in business for 20 months.

Explanation:

Part (a)

Step 1: Identify Initial Revenue

The initial revenue occurs at \( x = 0 \) (time = 0 months). On the graph, the \( y \)-intercept (where \( x = 0 \)) is at \( y = 20 \). So the initial revenue is the \( y \)-intercept of the graph.

Step 2: Determine the Value

From the graph, when \( x = 0 \), the revenue (in thousands of dollars) is 20. So the initial monthly revenue is 20 thousand dollars.

Step 1: Analyze the Graph's Trend

Looking at the graph for the first 10 months (from \( x = 0 \) to \( x = 10 \)), the curve is increasing (going up) but approaching a horizontal line (asymptote). So as time ( \( x \)) increases, the monthly revenue increases (but at a decreasing rate, approaching the asymptote).

Step 2: Conclusion on Trend

For the first 10 months, as time increases, the monthly revenue increases (towards the asymptote \( y = 100 \)).

Step 1: Identify the Asymptote

The graph approaches a horizontal line (horizontal asymptote) at \( y = 100 \). This is visible from the dashed line at \( y = 100 \) in the graph.

Step 2: Interpret the Asymptote

The horizontal asymptote \( y = 100 \) means the revenue approaches 100 thousand dollars but does not go above it. So the equation of the asymptote is \( y = 100 \), and the correct statement is "The company’s monthly revenue does not rise above 100 thousand dollars per month."

Answer:

20 (and the initial revenue is represented by the \( y \)-intercept of the graph)

Part (b)