Question

the graph shows the speed at which adourian was riding his bike during his 60 minute bike ride.
speed (miles per hour) vs. time (minutes)
during which of the following intervals did his speed change most rapidly?
a between (28,11) and (35,7)
b between (50,7) and (55,6)
c between (15,8) and (22,11)
d between (5,6) and (9,8)

Explanation:

To determine when the speed changed most rapidly, we calculate the rate of change (slope) for each interval. The slope formula is $\frac{\Delta y}{\Delta x}$, where $\Delta y$ is the change in speed (miles per hour) and $\Delta x$ is the change in time (minutes, converted to hours by dividing by 60).

Step 1: Analyze Option A

Interval: (30, 11) to (35, 7)
$\Delta y = 7 - 11 = -4$ mph
$\Delta x = 35 - 30 = 5$ minutes $= \frac{5}{60} = \frac{1}{12}$ hours
Slope (rate of change) = $\frac{-4}{\frac{1}{12}} = -48$ mph per hour (magnitude 48)

Step 2: Analyze Option B

Interval: (50, 7) to (55, 6)
$\Delta y = 6 - 7 = -1$ mph
$\Delta x = 55 - 50 = 5$ minutes $= \frac{5}{60} = \frac{1}{12}$ hours
Slope = $\frac{-1}{\frac{1}{12}} = -12$ mph per hour (magnitude 12)

Step 3: Analyze Option C

Interval: (15, 8) to (22, 11)
$\Delta y = 11 - 8 = 3$ mph
$\Delta x = 22 - 15 = 7$ minutes $= \frac{7}{60}$ hours
Slope = $\frac{3}{\frac{7}{60}} \approx 25.71$ mph per hour (magnitude ~25.71)

Step 4: Analyze Option D

Interval: (5, 6) to (9, 8)
$\Delta y = 8 - 6 = 2$ mph
$\Delta x = 9 - 5 = 4$ minutes $= \frac{4}{60} = \frac{1}{15}$ hours
Slope = $\frac{2}{\frac{1}{15}} = 30$ mph per hour (magnitude 30)

Answer:

The interval with the largest magnitude of slope (most rapid change) is Option A. So the answer is A. Between (30, 11) and (35, 7)