A local zoo found that if the price of admission was $17, the attendance was about 950 customers per week. When the price of admission was dropped to $8, attendance increased to about 2250 per week. Write a linear equation for the attendance in terms of the price, p.  ( A=mp+b )

We are tasked with finding the linear equation $ A = mp + b $, where $ A $ represents the attendance and $ p $ represents the price of admission.

We are given two points from the problem:
- When the price $ p = 17 $, the attendance $ A = 950 $.
- When the price $ p = 8 $, the attendance $ A = 2250 $.

### Step 1: Find the slope $ m $
To find the slope $ m $, we use the slope formula:
\[
m = \frac{\Delta A}{\Delta p} = \frac{A_2 - A_1}{p_2 - p_1}
\]
Substitute the given values:
\[
m = \frac{2250 - 950}{8 - 17} = \frac{1300}{-9} = -\frac{1300}{9}
\]

So, the slope is $ m = -\frac{1300}{9} $.

### Step 2: Find the y-intercept $ b $
To find $ b $, we use the point-slope form of the line $ A = mp + b $. Let's use one of the given points, say $ (p_1, A_1) = (17, 950) $.

Substitute $ m = -\frac{1300}{9} $, $ p = 17 $, and $ A = 950 $ into the equation:
\[
950 = -\frac{1300}{9} \times 17 + b
\]

Now, solve for $ b $:
\[
950 = -\frac{1300 \times 17}{9} + b
\]
\[
950 = -\frac{22100}{9} + b
\]
Convert $ 950 $ to a fraction with a denominator of 9:
\[
950 = \frac{8550}{9}
\]
Now solve for $ b $:
\[
\frac{8550}{9} = -\frac{22100}{9} + b
\]
\[
b = \frac{8550 + 22100}{9}
\]
\[
b = \frac{30650}{9}
\]
Thus, $ b = \frac{30650}{9} $, which is approximately 3405.56.

### Step 3: Write the linear equation
The linear equation for the attendance in terms of the price is:
\[
A = -\frac{1300}{9}p + \frac{30650}{9}
\]
or approximately:
\[
A = -144.44p + 3405.56
\]

This equation models the attendance as a function of the admission price.

Would you like more details on any of the steps or another aspect of this problem?

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Here are 5 related questions to explore this topic further:
1. What would the attendance be if the price was set at $12?
2. How would you interpret the slope in this problem context?
3. What would happen to the attendance if the admission was free (i.e., $ p = 0 $)?
4. How does changing the price by $1 affect the attendance?
5. Could the attendance ever drop to zero based on this model?

**Tip**: When solving for intercepts in linear models, remember to always isolate the unknown variable step by step to avoid calculation errors.

A local zoo found that if the price of admission was $17, the attendance was about 950 customers per week. When the price of admission was dropped to $8, attendance increased to about 2250 per week. Write a linear equation for the attendance in terms of the price, p.

Learn how to derive a linear equation that models zoo attendance as a function of admission price. When the price was $17, attendance was 950 customers per week, and at $8, it increased to 2250 per week. Explore the algebraic steps in calculating the slope and y-intercept, using real-world data. Suitable for students in Grades 9-11.

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