Question

an apple falls from a high tree branch. the height, h(t), in feet, of the apple above the ground is a function of the number of seconds, t, since the apple started falling. the table represents the function for selected values of t.

t	h(t)
0.2	15.36
0.4	13.44
0.5	12.00
0.6	10.24
0.8	5.76
1.0	0.00

which statement provides a correct interpretation of the average rate of change of this function from t = 0.5 to t = 1.0?

• the average rate of change is -12, which means that the apple fell at a constant speed of 12 feet per second.
• the average rate of change is -24, which means that the apple fell at a constant speed of 24 feet per second.
• the average rate of change is -12, which means that the apple traveled the same distance as something moving at a constant speed of 12 feet per second.
• the average rate of change is -24, which means that the apple traveled the same distance as something moving at a constant speed of 24 feet per second.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function \( h(t) \) from \( t = a \) to \( t = b \) is given by \( \frac{h(b)-h(a)}{b - a} \).

Step2: Identify values for \( t = 0.5 \) and \( t = 1.0 \)

From the table, when \( t = 0.5 \), \( h(0.5)=12.00 \) and when \( t = 1.0 \), \( h(1.0)=0.00 \).

Step3: Calculate average rate of change

Substitute \( a = 0.5 \), \( b = 1.0 \), \( h(a)=12.00 \), \( h(b)=0.00 \) into the formula:
\[
\frac{h(1.0)-h(0.5)}{1.0 - 0.5}=\frac{0.00 - 12.00}{0.5}=\frac{- 12.00}{0.5}=- 24
\]
The negative sign indicates the height is decreasing (the apple is falling). The average rate of change of -24 means the apple's height change over time is equivalent to an object moving at a constant speed of 24 feet per second (since it's the average rate, not necessarily constant speed, but the distance traveled on average is like that speed).

Answer:

The average rate of change is -24, which means that the apple traveled the same distance as something moving at a constant speed of 24 feet per second. (The option corresponding to this statement)