Question

A company can sell 2000 magazine subscriptions at $40 each. For each $5 increase in the price, it will sell 200 fewer subscriptions. What subscription price will provide the maximum revenue for the company?

Answer 1

Let x denote the number of times at which price is increased by $5 for each magazine.
{Write an expression for the total revenue of company}
Let y denote the total revenue of the company,
y = (2000 - 200x) × ($40 + $5x)
{200x is the number of subscriptions being lost due to the price increases}
{$5x is the amount of increase in price of each magazine}

y = 2000($40) + 2000($5x) - 200x($40) - 200x($5x)
y = $80,000 + $10,000x - $8000x - $1000x^2
y = $80,000 + $18'000x - $1000x^2
{Since the coefficient of x^2 is (-1000), this graph is a sad face curve. This means the only stationary point on the curve is a maximum point}
{Differentiate y in terms of x}
dy/dx = $18,000 - $2000x
{dy/dx = 0 at stationary point, or when gradient = 0}
dy/dx = 0
$18,000 - $2000x = 0
$2000x = $18,000
x = 9 {dollar signs cancel out when they are divided}

Subscription price to attain maximum revenue should be: $40 + (9 × $5) = $95

Hope this helps!