On a certain route, an aitine camies 7000 passengers per month, ench paying \( \$ 000 \). A market survey indicates that for each \( \$ \) decrease in the ticket price. the autline will gain 30 passengers a. Expess the number of passengers por month, \( N \), as a function of the ticket price, \( x \) b. Express the monthly revenue for the route, \( R \), as a function of the ticket price, \( x \) a. \( N(x)=13000-30 x \) b. \( R(x)=\square \)

Let's break this down in an engaging way! To express the monthly revenue \( R \) as a function of the ticket price \( x \), we first note that the number of passengers \( N \) can be represented as \( N(x) = 7000 + 30(7000 - x) \), where \( x \) is the price decrease from the base price of $100. Now, the revenue \( R \) can be calculated using the formula \( R = N \times \text{price per ticket} \). Hence, if we define the ticket price as \( 100 - x \), the revenue function becomes: \[ R(x) = N(x) \cdot (100 - x) \] So, plugging in \( N(x) \), we'd get: \[ R(x) = (7000 + 30x)(100 - x) \] Now, about the common pitfalls: many people forget to adjust the base number of passengers correctly when calculating changes in revenue based on ticket price adjustments. Always ensure you have a proper understanding of how your variables interact to avoid calculation errors!