Question

option 2: 5 marks
jaxon sells subscriptions for mathflix, a streaming service which provides math videos to help struggling students, for $8.10 per month. at this price, there are 300 subscribers each month. for every $.05 decrease in the price, 30 more students will purchase a subscription. what price should be charged to produce maximum revenue?

Explanation:

Step1: Define variables

Let \( x \) be the number of \(\$0.05\) decreases in price.
The price per subscription \( p=(8.10 - 0.05x)\) dollars.
The number of subscribers \( n=(300 + 30x)\)

Step2: Revenue function

Revenue \( R = p\times n=(8.10 - 0.05x)(300 + 30x)\)
Expand the function:

$$
LATEXBLOCK0
$$

Step3: Find the vertex of the quadratic function

For a quadratic function \(y = ax^{2}+bx + c\) (\(a=-1.5\), \(b = 228\), \(c = 2430\)), the \(x\) - coordinate of the vertex is \(x=-\frac{b}{2a}\)

$$ x=-\frac{228}{2\times(-1.5)}=\frac{228}{3}=76 $$

Step4: Calculate the price

Substitute \(x = 76\) into the price formula \(p=(8.10 - 0.05x)\)

$$ p=8.10-0.05\times76=8.10 - 3.80=4.3 $$

Answer:

The price that should be charged to produce maximum revenue is \(\$4.30\)