One of the most commonly used mathematical models for a demand function in micro-economics is $x=M p^{-k},$ where $p$ is the price of a commodity and $x$ is the quantity of the commodity that can be sold. Suppose that $M=4, k=$ $1 / 2, p$ is the price per ride on a New York City subway, and $x$ is the number of riders per day (in millions). (a) Find the ridership when the price is two dollars. (b) Find the drop in ridership if the price is raised to 2.25 dollar. (c) Find the equation of the tangent line to the demand curve at $p=2 .$ (d) Use the value of $x$ on the straight line to find the approximate drop in ridership if the price is raised to 2.25 dollar. This is called the marginal demand. (e) Find the ratio of the value found in (d) to the change in price.