1. ## Simultaneous Equations

A journal is planning to increase the price of its magazine. According to a recent survey, 1250 magazines will decrease for each 0.025 dollars increase of the magazine price. What is the price of the magazine that will increase the revenue of the journal to its maximum. The normal price of the magazine is 7.5 dollars, and the normal number of magazines sold is 500,000.

2. ## Re: Simultaneous Equations

let $i$ be the increase in price

$p = 7.5 + i$

the number of magazines sold will be

$n = 500000 - \dfrac{i}{0.025}\cdot 1250$

the problem doesn't clearly state if this is supposed to be linear or stepwise. We'll assume linear.

the revenue will be

$r = n p = (7.5+i)\left(500000 - \dfrac{i}{0.025}\cdot 1250\right) = -50000(i^2 -2.5i-75)$

This is a downward facing parabola with a maximum at the vertex. So let's get the parabola in standard form by completing the square.

\begin{align*} &i^2 - 2.5i - 75 = \\ \\ &(i-1.25)^2 - (1.25)^2 - 75 = \\ \\ &(i-1.25)^2 - 76.5625 \end{align*}

and then multiplying by the $-50000$ we get

$r = 3878125 - 50000(i-1.25)^2$

and we can read off that the best price increase is $i=1.25$

3. ## Re: Simultaneous Equations

Thanks a lot. I got the idea.