Questions: 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)

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)

Transcript text: 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, $p .(A=mp+b)$

Solution

Solution Steps

Step 1: Identify the given points

We are given two points:

When the price of admission was $17, the attendance was 950 customers per week. This gives us the point (17, 950).
When the price of admission was $8, the attendance was 2250 customers per week. This gives us the point (8, 2250).

Step 2: Calculate the slope (m)

The slope $ m $ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the given points: \[ m = \frac{2250 - 950}{8 - 17} = \frac{1300}{-9} = -\frac{1300}{9} \approx -144.44 \]

Step 3: Use the point-slope form to find the equation

The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \]

Using the point (17, 950) and the slope $ m = -\frac{1300}{9} $: \[ y - 950 = -\frac{1300}{9}(x - 17) \]

Step 4: Simplify to get the linear equation

Distribute the slope and simplify: \[ y - 950 = -\frac{1300}{9}x + \frac{1300 \times 17}{9} \] \[ y - 950 = -\frac{1300}{9}x + \frac{22100}{9} \] \[ y = -\frac{1300}{9}x + \frac{22100}{9} + 950 \] \[ y = -\frac{1300}{9}x + \frac{22100}{9} + \frac{8550}{9} \] \[ y = -\frac{1300}{9}x + \frac{30650}{9} \]