Question

WHAT IF? A streaming service company charges $8 per month and has 15 million subscribers. For each $1 increase in price, the company loses 1.5 million subscribers. How much should the company charge to maximize monthly revenue?

Answer 1

So, to maximize monthly revenue, the company should charge $9 per month, and the maximum revenue will be $121.5 million.

To maximize monthly revenue, we need to find the price at which the revenue is maximized. Revenue is calculated by multiplying the price per subscriber by the number of subscribers.

Let P be the price increase in dollars, and Q be the number of subscribers.

The original price is $8, and the original number of subscribers is 15 million.

So, the original revenue
`(\( R_0 \))` is given by:

`\[ R_0 = P_0 * Q_0 = 8 * 15 \, \text{million} \]`

Now, for each $1 increase in price, the company loses 1.5 million subscribers. So, if the price increases by P dollars, the number of subscribers Q will be
`\( 15 - 1.5P \)` million.

The revenue R with the price increase is given by:

`\[ R = (8 + P) * (15 - 1.5P) \, \text{million} \]`

Now, we want to find the value of P that maximizes R. To do this, we can find the critical points by taking the derivative of R with respect to P and setting it equal to zero:

`\[ (dR)/(dP) = 0 \]`

`\[ (d)/(dP) [(8 + P) * (15 - 1.5P)] = 0 \]`

`\[ (15 - 1.5P) - (8 + P) * 1.5 = 0 \]`

Sure, let's solve for P in the equation:

`\[ (15 - 1.5P) - (8 + P) * 1.5 = 0 \]`

First, distribute
`\( 1.5 \)` on the right side:

`\[ 15 - 1.5P - 12 - 1.5P = 0 \]`

Combine like terms:

`\[ 3 - 3P = 0 \]`

Now, isolate
`\( P \)`:

`\[ -3P = -3 \]`

Divide both sides by
`\( -3 \)`:

`\[ P = 1 \]`

So,
`\( P = 1 \)` is the solution. Now, we need to determine the corresponding price and calculate the revenue.

The original price is $8, and for each $1 increase, the new price is
`\( 8 + P = 8 + 1 = 9 \)` dollars.

Now, calculate the revenue
`(\( R \))`:

`\[ R = (8 + P) * (15 - 1.5P) \]`

`\[ R = 9 * (15 - 1.5 * 1) \]`

`\[ R = 9 * 13.5 \]`

`\[ R = 121.5 \, \text{million dollars} \]`