Question

1. a roller skating rink charges a skate rental fee and an hourly rate to skate. the total cost to skate for 2 hours is $9.50 and for 5 hours is $18.50. assume the relationship is linear, where x represents the number of hours and y represents the total cost. find and interpret the rate of change and initial value. (example 4)

Explanation:

Step1: Find the rate of change (slope)

The rate of change (slope) \(m\) is calculated using the formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(2,9.50)\) and \((x_2,y_2)=(5,18.50)\). Then \(m=\frac{18.50 - 9.50}{5 - 2}=\frac{9}{3}=3\).

Step2: Find the initial value (y - intercept)

Use the point - slope form \(y - y_1=m(x - x_1)\). Substitute \(m = 3\), \(x_1=2\), \(y_1 = 9.50\). \(y-9.50=3(x - 2)\). Expand: \(y-9.50=3x-6\). Then \(y=3x + 3.50\). The initial value (when \(x = 0\)) is \(y=3.50\).

Answer:

The rate of change (slope) is \(3\) (meaning it costs \( \$3\) per hour to skate after paying the rental fee) and the initial value (y - intercept) is \(3.50\) (the skate rental fee).