Question

a movie theater offers a reward program that charges a yearly membership fee and a discounted rate per movie ticket. the total cost for a reward program member to see 5 movies is $40 and the total cost for 12 movies is $75. assume the relationship is linear, where x represents the number of movies and y represents the total cost. find and interpret the rate of change and initial value. the rate of change is \boxed{}, so the cost per ticket is $ \boxed{}. the initial value is \boxed{}, so the initial membership fee is $ \boxed{}.

Explanation:

Step1: Define the linear equation

Let the linear equation be \( y = mx + b \), where \( y \) is total cost, \( x \) is number of movies, \( m \) is rate of change (cost per movie), and \( b \) is initial value (membership fee). We know two points: when \( x = 5 \), \( y = 40 \); when \( x = 12 \), \( y = 75 \).

Step2: Calculate the rate of change (slope)

The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substituting \( (x_1,y_1)=(5,40) \) and \( (x_2,y_2)=(12,75) \), we get \( m=\frac{75 - 40}{12 - 5}=\frac{35}{7} = 5 \). So the rate of change (cost per ticket) is $5.

Step3: Find the initial value (membership fee)

Use the point \( (x = 5, y = 40) \) and \( m = 5 \) in \( y = mx + b \). So \( 40=5\times5 + b \). Simplify: \( 40 = 25 + b \). Subtract 25 from both sides: \( b = 40 - 25 = 15 \). So the initial membership fee (initial value) is $15.

Answer:

The rate of change (cost per ticket) is \$5, the initial value (initial membership fee) is \$15.