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:
time (seconds), x | distance (yards), y
0 | 0
10 | 70
20 | 140
what is paula’s speed? find the rate of change.
paula’s speed = 7 yards per second
what is nancy’s speed? find the rate of change.
nancy’s speed = □ yards per second

Explanation:

Step1: Recall Speed Formula

Speed is calculated as $\frac{\text{Distance}}{\text{Time}}$. For Nancy, we know she runs 6 yards per second? Wait, no, wait. Wait, the problem says "Nancy runs 6 yards per second"? Wait, no, let's check again. Wait, the question is about Nancy's speed. Wait, maybe I misread. Wait, the problem: "Nancy is new to the team, so she starts 10 yards ahead of the starting line. Nancy runs 6 yards per second." Wait, no, maybe the question is to confirm? Wait, no, the sub - question is "What is Nancy’s speed? Find the rate of change." Wait, the problem statement says "Nancy runs 6 yards per second." Wait, but maybe from the context, but let's check. Wait, the speed is the rate of change of distance with respect to time. For a linear function (since speed is constant), the rate of change (slope) is the speed. If Nancy's motion is modeled as a function, and she runs at a constant speed, and the problem states "Nancy runs 6 yards per second", so her speed is 6 yards per second. Wait, but let's confirm. The rate of change for distance - time function (when speed is constant) is speed, which is $\frac{\Delta y}{\Delta x}$ where $y$ is distance and $x$ is time. If Nancy starts 10 yards ahead (initial value 10) and runs at a constant speed, then her distance function is $y = 6x+10$, so the rate of change (slope) is 6, which is her speed.

Step2: Determine Nancy's Speed

From the problem description, it is stated that "Nancy runs 6 yards per second". Also, from the concept of speed as the rate of change of distance with respect to time, for a linear motion with constant speed, the speed is the coefficient of $x$ in the distance - time function $y=mx + b$, where $m$ is speed (rate of change) and $b$ is initial distance. Since Nancy's speed is given (or can be inferred as 6 yards per second from the problem context, as the rate of change of her distance with respect to time is 6 yards per second).

Answer:

6