1. The monthly revenue, $\mathrm{R}(\mathrm{x})$, for a big tech company depends on the monthly advertising expenses, $x$, by the function $R(x)=\frac{1020 x}{25+8 x}$, where the monthly revenue and advertising expenses are in thousands of dollars.
(a) What should be the monthly advertising expenses if the company wants the monthly revenue (in $\$$ ) to exceed \$200000? Include an interval chart in your solution. (3 marks)
(b) What will happen to the amount of monthly revenue as advertising expenses increase? Provide a complete answer by providing details about the behaviour of the graph and numerical details about the monthly revenue. Include a sketch. (2 marks)

Step 1 ：Let's denote the monthly revenue as \(R(x)\) and the monthly advertising expenses as \(x\). The relationship between them is given by the function \(R(x) = \frac{1020x}{25 + 8x}\), where both \(R(x)\) and \(x\) are in thousands of dollars.

Step 2 ：To find the monthly advertising expenses that would make the monthly revenue exceed $200,000, we need to solve the inequality \(R(x) > 200\).

Step 3 ：Solving the inequality, we get \(x > -\frac{250}{29}\). However, since advertising expenses cannot be negative, the practical solution is \(x > 0\).

Step 4 ：\(\boxed{x > 0}\) is the solution to the inequality, meaning that the monthly advertising expenses should be more than $0 for the monthly revenue to exceed $200,000.

Step 5 ：To analyze the behavior of the function \(R(x)\) as \(x\) increases, we can plot the function and observe its behavior.

Step 6 ：From the plot, we can observe that as advertising expenses increase, the monthly revenue also increases. However, the rate of increase of the monthly revenue decreases as advertising expenses increase. This is indicated by the flattening of the curve as \(x\) increases.

Step 7 ：\(\boxed{\text{As advertising expenses increase, the monthly revenue also increases. However, the rate of increase of the monthly revenue decreases as advertising expenses increase.}}\)