Linear approximation of bus revenue

In summary, the conversation discusses the relationship between the price of a bus pass and the monthly revenue of a bus company. A formula, R(x) = 1.5x − 0.01x2, is provided to calculate the revenue in thousands of dollars based on the price, x. The conversation then considers the effects of a small increase in price and uses the Linear Approximation to explain how it would result in a slight decrease in revenue. The Linear Approximation is compared to the equation y=mx+b to aid in understanding.
  • #1
bcahmel
25
0

Homework Statement


If the price of a bus pass from Albuquerque to Los Alamos is set at x dollars, a bus company takes in a monthly revenue of R(x) = 1.5x − 0.01x2 (in thousands of dollars).

Suppose that x = 80. How will revenue be affected by a small increase in price? Explain using the Linear Approximation.


The Attempt at a Solution


first I took the derivative which is 1.5-0.02x. Then I plugged 80 in for x, so f'(x)=-0.1. Is this right? So the revenue will slightly decrease.
 
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  • #2
so the linear approximation is
[tex] R(x) \approx R(x_0) +R'(x_0)(x - x_0) [/tex]
 
  • #3
ok, so its just like y=mx+b, sort of..
 
  • #4
Which is why they call it a linear approximation...
 
  • #5
yes, I get it now...thanks mark44
 

1. What is linear approximation of bus revenue?

Linear approximation of bus revenue is a method used to estimate the total revenue generated by a bus company based on a linear relationship between the number of passengers and the fare price.

2. How is linear approximation of bus revenue calculated?

The linear approximation of bus revenue is calculated by first determining the slope of the linear relationship between the number of passengers and the fare price. This can be done by finding the average fare price and the average number of passengers, and then using the formula for slope (change in y divided by change in x). Once the slope is determined, it can be multiplied by the number of passengers to estimate the total revenue.

3. What are the assumptions made in linear approximation of bus revenue?

The main assumption in linear approximation of bus revenue is that the relationship between the number of passengers and the fare price is linear, meaning that as one variable increases, the other also increases at a constant rate. This method also assumes that there are no external factors, such as weather or events, that may affect the number of passengers or the fare price.

4. Is linear approximation of bus revenue an accurate method?

Linear approximation of bus revenue is a simplified method and may not be entirely accurate. It provides an estimate based on the assumption of a linear relationship, but it may not take into account other factors that may affect the actual revenue. Therefore, it should be used as a rough estimate and not as a precise calculation of bus revenue.

5. How is linear approximation of bus revenue useful in real-life situations?

Linear approximation of bus revenue can be useful for bus companies to estimate their potential revenue based on the expected number of passengers. It can also be used to make projections and plan for future investments or expansions. Additionally, it can be helpful for passengers to understand the relationship between fare prices and the number of passengers, and how changes in one may affect the other.

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