A volume discount on a certain item is expressed as a differential equation in which the derivative of the price per item with respect to the number purchased is proportional to the difference between the price and some base price, below which the price can never go. If the price for just one item is $60, and the base price is $30, and the price per item when buying 10 items is $54, what is the price per item when buying 100 items?
A) $33
B) $48
C) $57
D) $46

Question

Quizplus · Accepted Answer

The given differential equation can be modeled as $\frac{dp}{dn} = k(p - 30)$, where $p$ is the price per item, $n$ is the number of items, and $k$ is a constant of proportionality. This is a separable differential equation, and its solution follows an exponential decay model towards the base price of $30. Given the initial condition $p(1) = 60$ and $p(10) = 54$, we can find $k$ and then use it to find $p(100)$. The solution shows that as $n$ increases, $p(n)$ approaches $30, but does so gradually. Given the options and the nature of exponential decay, the price per item when buying 100 items would be closest to the base price but not equal to it, making $33 the correct choice, as it is the only option slightly above the base price of $30.

A Volume Discount on a Certain Item Is Expressed as a Differential