Question

paula and nancy race at track practice. paula records her time in a table where her distance in yards, y, is a function of her time in seconds, x. nancy is new to the team, so she starts 10 yards ahead of the starting line. nancy runs 6 yards per second.
initial value for paula’s function: 0
initial value for nancy’s function: 10
paula’s rate of change: 7
nancy’s rate of change: 6
compare the speeds of the runners.
the rate of change for paula is? the rate of change for nancy. that means? runs at a faster speed.
paula:

time (seconds), x	0	10	20
distance (yards), y	0	70	140

Explanation:

Step1: Calculate Paula's rate of change

The rate of change (slope) formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(10,70)\) from Paula's table:
\(m=\frac{70 - 0}{10 - 0}=\frac{70}{10}=7\)

Step2: Compare rates of change

Paula's rate of change \(m_{Paula}=7\) and Nancy's rate of change \(m_{Nancy}=6\) (given as she runs 6 yards per second). Since \(7>6\)

Answer:

The rate of change for Paula is greater than the rate of change for Nancy. That means Paula runs at a faster speed.