The number of daily viewers for a new presenter on a video gaming streaming service can be
modelled by the following equation:
V = –11t

2 + 528t {0 ≤ t ≤ 48}

Where V represents the number of daily viewers and t represents time in months.
(i) Find the rate(s) of change of daily viewers, with respect to t, when the number of daily
viewers is 3520.

We are given the equation for the number of daily viewers \( V \) in terms of time \( t \) in months:

\[
V = -11t^2 + 528t \quad \text{for} \quad 0 \leq t \leq 48
\]

### (i) Finding the rate(s) of change of daily viewers when \( V = 3520 \):

1. First, differentiate the equation for \( V \) with respect to \( t \) to find the rate of change of daily viewers, which is the derivative \( \frac{dV}{dt} \):

\[
\frac{dV}{dt} = \frac{d}{dt} \left( -11t^2 + 528t \right)
\]
Applying the power rule to differentiate each term:

\[
\frac{dV}{dt} = -22t + 528
\]

This represents the rate of change of daily viewers at any time \( t \).

2. Next, we need to find when the number of daily viewers is 3520. So, we solve the equation \( V = 3520 \) for \( t \):

\[
-11t^2 + 528t = 3520
\]
Rearranging this equation:

\[
-11t^2 + 528t - 3520 = 0
\]

This is a quadratic equation. We will solve for \( t \) using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = -11 \), \( b = 528 \), and \( c = -3520 \).

Let me calculate the values of \( t \).The two possible values for \( t \) are:

\[
t = 8 \quad \text{or} \quad t = 40
\]

These are the times (in months) when the number of daily viewers is 3520.

3. Now, we substitute these values of \( t \) into the derivative \( \frac{dV}{dt} = -22t + 528 \) to find the rate of change of daily viewers at those times:

- When \( t = 8 \):

\[
\frac{dV}{dt} = -22(8) + 528
\]

- When \( t = 40 \):

\[
\frac{dV}{dt} = -22(40) + 528
\]

Let me calculate these values for the rate of change.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can still help guide you through the final steps of the problem.

To find the rate of change at \( t = 8 \) and \( t = 40 \), substitute those values into the derivative:

For \( t = 8 \):
\[
\frac{dV}{dt} = -22(8) + 528 = 352 \quad \text{viewers per month}
\]

For \( t = 40 \):
\[
\frac{dV}{dt} = -22(40) + 528 = -352 \quad \text{viewers per month}
\]

Thus, at \( t = 8 \), the daily viewers are increasing at a rate of 352 viewers per month, and at \( t = 40 \), they are decreasing at a rate of 352 viewers per month.

Would you like more details or have any questions? Here are five related questions to consider:

1. How does the number of viewers change over time according to this model?
2. What is the maximum number of daily viewers, and when does it occur?
3. How would the graph of \( V(t) \) look based on this equation?
4. What happens to the number of viewers as \( t \) approaches 48 months?
5. How would this model change if the presenter gained more viewers at the start?

**Tip:** When solving quadratic equations, always check for multiple solutions, especially in real-world scenarios like this!

The number of daily viewers for a new presenter on a video gaming streaming service can be modelled by the following equation: V = –11t^2 + 528t {0 ≤ t ≤ 48} Where V represents the number of daily viewers and t represents time in months. (i) Find the rate(s) of change of daily viewers, with respect to t, when the number of daily viewers is 3520.

Learn how to find the rate of change of daily viewers for a gaming presenter using a quadratic equation. The problem involves differentiation and solving for time using the quadratic formula. Suitable for high school students in Grades 11-12.

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