Question

A rubber ball dropped on a hard surface takes a sequence of bounces, each one 4/5 as high as the preceding one. If this ball is dropped from a height of 10 feet, what is the total vertical distance it has traveled after it hits the surface the fifth time? Be sure to account for both drop and rebound distances.

We are learning sequences and series I'm confused on how to do this >.<

Answer 1

I just learned this in my class, I am pretty sure I can help you, let me go find my notes on it.

Answer 2

Ok :)

Answer 3

This is a hybrid G.P. with T1=8, r=0.8, n=4, (excluding the first fall of 10 feet).

i.e. to say:
Total vertical distance = [2x(Sum of G.P. where T1=8, r=0.8, n=4)]+10 feet.

Answer 4

ok so if the height is 10 and the rebound is 4/5 we want to first find the ratio between the sequences

Answer 5

\[10(ratio)^{i-1}\sum_{i-1}^{5} \frac{ 4 }{ 5 }(ratio)^{i-1}\]would be the formula set up we would want to use after we found the ratio

Answer 6

Wouldn't the ratio be 4/5 ? "4/5 as high as the preceding one"

Answer 7

oh you are right, so 4/5 would go in the part where I have ratio, and 4/5 of 10 would go were I have 4/5

Answer 8

since we are not going to infinity and just five it would be i not i-1. so the rebound would be 4/5 or 8/10 the first bounce, so the equation would look like\[10(\frac{ 4 }{ 5 })^{i}\sum_{i}^{5}\frac{ 4 }{ 5}(\frac{ 4 }{ 5})^{i}\]

Answer 9

\[10(\frac{ 1 }{ 1-\frac{ 4 }{ 5 } })-5(\frac{ 1 }{ 1-\frac{ 4 }{ 5 } })\]

Answer 10

50-15=45 ft

Answer 11

does that look right?

Answer 12

Not really? I'm confused to be honest >.<

Answer 13

k what do you understand, and I'll pick up from there

Answer 14

Hold on a second let me reread your stuff :)

Answer 15

Okay doesn't
\[\sum_{n=1}^5 10 (\frac{4}{5})^n \]
that work?

Answer 16

*why doesn't that work?

Answer 17

That can, I used it so that we had a1 and a2, but if we wanted to work with just a1 that is probably a simpler approach. I think I worked it out to how far the ball traveled by the time it came to a stop anyways, and not the fifth bounce. Looking back over my notes I think it would be easier to use the formula\[S _{n}=a _{1}\frac{ 1-r ^{n} }{ 1-r }\] where S=total distance traveled r=10 and n=5

Answer 18

well a= 10 and r= 4/5

Answer 19

That gives me:
33.616 + 33.616 = 67.232
Which isn't one of my answer choices :/

Answer 20

\[S _{5}=10\frac{ 1-\frac{ 4 }{ 5 }^{5} }{ 1-\frac{ 4 }{ 5 } }\]

Answer 21

what are your answer choices?

Answer 22

A 28 77/125
B 49 1/25
C 57 29/125
D 59 29/125

Answer 23

well we could always make a table, and go about it without an equation

Answer 24

Uh not sure what I exactly would do with a table :/ explain please?

Answer 25

well each time the ball drops it comes back up 4/5 distance, so it drops ten, comes up 4/5 which means it didn't not come back up 1/5 of the distance so 10-1/5 is 49/5. The table shows each distance of fall, with each distance of rebound. Then we add them all together to get the answer.

Answer 26

I think the measurements on the table are a little off. Let me try to convert them to decimals, see if I got it wrong some where.

Answer 27

nope i keep getting 86, so I'm not sure. THe formula Shiraz14 gave doesn't make sense to me either. I do know... I thought I could figure it out, but obviously I need more practice with these problems. Sorry :(