Question

deshawna and her team gathered data for their ball and recorded it in the table shown at right.
a. what is the rebound ratio for their ball?
b. predict how high deshawnas ball will rebound if it is dropped from 275 cm. look at the precision of deshawnas measurements in the table. round your calculation to a reasonable number of decimal places.
c. suppose the ball is dropped and you notice that its rebound height is 60 cm. from what height was the ball dropped? use an appropriate precision for your answer.
d. suppose the ball is dropped from a window 200 meters up the empire state building. what would you predict the rebound height to be after the first bounce?
e. how high would the ball in part (d) rebound after the second bounce? after the third bounce?

Explanation:

Step1: Calculate rebound ratio

Rebound ratio \(r=\frac{\text{Rebound Height}}{\text{Drop Height}}\).
For \(150\) cm drop and \(124\) cm rebound: \(r_1 = \frac{124}{150}\approx0.827\)
For \(70\) cm drop and \(59\) cm rebound: \(r_2=\frac{59}{70}\approx0.843\)
For \(120\) cm drop and \(100\) cm rebound: \(r_3=\frac{100}{120}\approx0.833\)
For \(100\) cm drop and \(83\) cm rebound: \(r_4=\frac{83}{100} = 0.83\)
For \(110\) cm drop and \(92\) cm rebound: \(r_5=\frac{92}{110}\approx0.836\)
For \(40\) cm drop and \(33\) cm rebound: \(r_6=\frac{33}{40}=0.825\)
Average \(r=\frac{0.827 + 0.843+0.833+0.83+0.836+0.825}{6}\)
\(=\frac{4.994}{6}\approx0.83\)

Step2: Predict rebound height for \(275\) cm drop (part b)

Using \(r = 0.83\), rebound height \(h=275\times0.83 = 228.25\approx228\) cm

Step3: Find drop height for \(60\) cm rebound (part c)

Let drop height be \(H\). Since \(r = 0.83\), \(H=\frac{60}{0.83}=\frac{6000}{83}\approx72.3\) cm

Step4: Predict rebound height for \(200\) m drop (part d)

First convert \(200\) m to \(20000\) cm. Rebound height \(h=20000\times0.83 = 16600\) cm or \(166\) m

Step5: Predict second - bounce height (part e)

Second - bounce height \(h_2=166\times0.83 = 137.78\approx138\) m
Third - bounce height \(h_3=138\times0.83 = 114.54\approx115\) m

Answer:

a. The rebound ratio is approximately \(0.83\).
b. The ball will rebound approximately \(228\) cm.
c. The ball was dropped from approximately \(72.3\) cm.
d. The rebound height after the first bounce is \(166\) m.
e. After the second bounce: approximately \(138\) m; after the third bounce: approximately \(115\) m.