Question

a tennis ball is tossed into the air. the height, h(t), in feet, of the tennis ball is a function of time, t, in seconds, as shown in the table below

t	h(t)
0.5	20
1	32
1.5	36
2	32
2.5	20
3	0

which statement provides a correct interpretation of the average rate of change of this function from t = 1.5 to t = 3?

a. the average rate of change is 24, which means that the tennis ball was rising at an average speed of 24 ft/sec.

b. the average rate of change is 24, which means that the tennis ball was falling at an average speed of 24 ft/sec.

c. the average rate of change is -24, which means that the tennis ball was rising at an average speed of 24 ft/sec.

d. the average rate of change is -24, which means that the tennis ball was falling at an average speed of 24 ft/sec.

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 \( \frac{h(b) - h(a)}{b - a} \). Here, \( a = 1.5 \), \( b = 3 \), \( h(1.5)=36 \), \( h(3)=0 \).

Step2: Calculate average rate of change

Substitute values into the formula: \( \frac{h(3)-h(1.5)}{3 - 1.5}=\frac{0 - 36}{1.5}=\frac{-36}{1.5}=-24 \).

Step3: Interpret the result

A negative average rate of change means the height is decreasing (falling) over time. The magnitude 24 means the average speed of falling is 24 ft/sec.

Answer:

D. The average rate of change is \(-24\), which means that the tennis ball was falling at an average speed of 24 ft/sec.