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Let's break down the problem step by step.

The given equation is:

\[
P(t) = -5t^2 + 75t - 30
\]

### 1. Find $ P'(t) $

To find the derivative $ P'(t) $, we'll differentiate the function with respect to $ t $.

\[
P'(t) = \frac{d}{dt}(-5t^2 + 75t - 30)
\]

Now I'll compute this derivative.The derivative of the function is:

\[
P'(t) = 75 - 10t
\]

### 2. Find $ P'(5) $

Now, let's calculate $ P'(5) $ by substituting $ t = 5 $ into the derivative.By substituting $ t = 5 $ into the derivative, we get:

\[
P'(5) = 25
\]

### Interpretation:
The value $ P'(5) = 25 $ means that at $ t = 5 $ (which is 5 years after the start of 2000, or the year 2005), the median home price was increasing at a rate of $25,000 per year.

Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How would the home price change if the derivative was negative?
2. What is the significance of $ P'(t) $ in terms of home prices over time?
3. How can we use the second derivative to understand the acceleration of price changes?
4. What happens to the home price trend after $ t = 5 $?
5. How would external economic factors affect the model for $ P(t) $?

**Tip**: The derivative $ P'(t) $ gives the rate of change of home prices, helping predict future trends or declines in pricing.

The median home price in the United States over the period 2003-2011 can be approximated by P(t) = -5t^2 + 75t - 30 thousand dollars (3 ≤ t ≤ 11), where t is time in years since the start of 2000. Find P'(t) and P'(5).

This problem involves calculating the derivative of a quadratic function representing median home prices over time. Learn how to find P'(t) and interpret the rate of change of home prices in the U.S. between 2003 and 2011, suitable for high school or college students.

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