Question

rate of change

1. the rate of change is constant in the graph.

descent of a skydiver
height after parachute opens (feet)
time (seconds)
a.) what is the rate of change?
b.) explain what the rate of change means.
c.) how fast does the skydiver descend in minutes?
d.) how fast does the skydiver descend in hours?

1. miguel tracked the weekly spread of strains of flu virus. his data are shown below. compare the number of cases of flu by finding and interpreting average rates of change over the interval 0, 4.

strain 1
week 0 1 2 3 4
cases 15 25 40 60 85
strain 2
which strain has the higher rate

Explanation:

Step1: Find rate of change for skydiver

The rate of change (slope) formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. From the skydiver - graph, when $x_1 = 0,y_1=2000$ and when $x_2 = 60,y_2 = 1000$. So $m=\frac{1000 - 2000}{60-0}=\frac{- 1000}{60}=-\frac{50}{3}\text{ feet per second}$.

Step2: Explain the rate of change

The rate of change $-\frac{50}{3}\text{ feet per second}$ means the skydiver's height is decreasing by $\frac{50}{3}$ feet every second.

Step3: Convert to feet per minute

Since there are 60 seconds in a minute, the speed in feet per minute is $-\frac{50}{3}\times60=-1000\text{ feet per minute}$.

Step4: Convert to feet per hour

Since there are 60 minutes in an hour, the speed in feet per hour is $-1000\times60=-60000\text{ feet per hour}$.

Step5: Find average rate of change for Strain 1

For Strain 1, using the formula for average rate of change $m=\frac{y_2 - y_1}{x_2 - x_1}$ over the interval $[0,4]$. Here $x_1 = 0,y_1 = 15$ and $x_2 = 4,y_2 = 85$. So $m=\frac{85 - 15}{4-0}=\frac{70}{4}=17.5$ cases per week.

Step6: Estimate average rate of change for Strain 2

From the graph of Strain 2, when $x = 0,y\approx10$ and when $x = 4,y\approx70$. The average rate of change $m=\frac{70 - 10}{4-0}=\frac{60}{4}=15$ cases per week. Since $17.5>15$, Strain 1 has a higher average rate of change over the interval $[0,4]$.

Answer:

a. The rate of change for the skydiver is $-\frac{50}{3}\text{ feet per second}$.
b. It means the skydiver's height is decreasing by $\frac{50}{3}$ feet every second.
c. The skydiver descends at a rate of 1000 feet per minute.
d. The skydiver descends at a rate of 60000 feet per hour.
For the flu - virus strains: Strain 1 has a higher average rate of change over the interval $[0,4]$.