Mixing
Convergence of geometric series factor 3/4 word problem
Clash Royale CLAN TAG#URR8PPP
up vote
-1
down vote
favorite
I came across this problem on AoPS: A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to three quarters its previous height. So, after its first bounce it rises to 75 ft, and after its second bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels?
I reduced it to a fairly simple geometric series:
$$
sumlimits_n = 0^infty 100(frac34)^n
$$
Using $ sumlimits_n = 0^infty ax^n = fraca1-x $ with a=100 and x=3/4, the answer I got was 400 but the answer key said the answer was 700. Did I read the problem wrong and the distance was actually supposed to be measured differently? I don't think the answer key is wrong since the rest of its answers checked out. Any help would be appreciated. Thx in advance.
P.S. Not really sure what tags to put on this so pls add a tag that will fit the question.
convergence word-problem geometric-series
asked Jul 23 at 12:58
Suhani Shukla
1
add a comment |Â
up vote
-1
down vote
favorite
I came across this problem on AoPS: A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to three quarters its previous height. So, after its first bounce it rises to 75 ft, and after its second bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels?
I reduced it to a fairly simple geometric series:
$$
sumlimits_n = 0^infty 100(frac34)^n
$$
Using $ sumlimits_n = 0^infty ax^n = fraca1-x $ with a=100 and x=3/4, the answer I got was 400 but the answer key said the answer was 700. Did I read the problem wrong and the distance was actually supposed to be measured differently? I don't think the answer key is wrong since the rest of its answers checked out. Any help would be appreciated. Thx in advance.
P.S. Not really sure what tags to put on this so pls add a tag that will fit the question.
convergence word-problem geometric-series
asked Jul 23 at 12:58
Suhani Shukla
1
1
It goes up and comes down on each bounce :)
– dbx
Jul 23 at 13:01
Just multiply $400$ by $2$ to count the up and down motion, and then subtract $100$ because the first bounce does not have a corresponding upwards motion (if that makes sense).
– RayDansh
Jul 23 at 13:05
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
I came across this problem on AoPS: A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to three quarters its previous height. So, after its first bounce it rises to 75 ft, and after its second bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels?
I reduced it to a fairly simple geometric series:
$$
sumlimits_n = 0^infty 100(frac34)^n
$$
Using $ sumlimits_n = 0^infty ax^n = fraca1-x $ with a=100 and x=3/4, the answer I got was 400 but the answer key said the answer was 700. Did I read the problem wrong and the distance was actually supposed to be measured differently? I don't think the answer key is wrong since the rest of its answers checked out. Any help would be appreciated. Thx in advance.
P.S. Not really sure what tags to put on this so pls add a tag that will fit the question.
convergence word-problem geometric-series
asked Jul 23 at 12:58
Suhani Shukla
1
I came across this problem on AoPS: A rubber ball is dropped from a 100 ft tall building. Each time it bounces, it rises to three quarters its previous height. So, after its first bounce it rises to 75 ft, and after its second bounce it rises to 3/4 of 75 ft, and so on forever. What is the total distance the ball travels?
I reduced it to a fairly simple geometric series:
$$
sumlimits_n = 0^infty 100(frac34)^n
$$
Using $ sumlimits_n = 0^infty ax^n = fraca1-x $ with a=100 and x=3/4, the answer I got was 400 but the answer key said the answer was 700. Did I read the problem wrong and the distance was actually supposed to be measured differently? I don't think the answer key is wrong since the rest of its answers checked out. Any help would be appreciated. Thx in advance.
P.S. Not really sure what tags to put on this so pls add a tag that will fit the question.
convergence word-problem geometric-series
asked Jul 23 at 12:58
Suhani Shukla
1
asked Jul 23 at 12:58
Suhani Shukla
1
asked Jul 23 at 12:58
Suhani Shukla
1
asked Jul 23 at 12:58
Suhani Shukla
1
1
1
It goes up and comes down on each bounce :)
– dbx
Jul 23 at 13:01
Just multiply $400$ by $2$ to count the up and down motion, and then subtract $100$ because the first bounce does not have a corresponding upwards motion (if that makes sense).
– RayDansh
Jul 23 at 13:05
add a comment |Â
1
It goes up and comes down on each bounce :)
– dbx
Jul 23 at 13:01
Just multiply $400$ by $2$ to count the up and down motion, and then subtract $100$ because the first bounce does not have a corresponding upwards motion (if that makes sense).
– RayDansh
Jul 23 at 13:05
1
1
It goes up and comes down on each bounce :)
– dbx
Jul 23 at 13:01
It goes up and comes down on each bounce :)
– dbx
Jul 23 at 13:01
Just multiply $400$ by $2$ to count the up and down motion, and then subtract $100$ because the first bounce does not have a corresponding upwards motion (if that makes sense).
– RayDansh
Jul 23 at 13:05
Just multiply $400$ by $2$ to count the up and down motion, and then subtract $100$ because the first bounce does not have a corresponding upwards motion (if that makes sense).
– RayDansh
Jul 23 at 13:05
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
You counted only the distance the ball traveled downward on each bounce. The answer key is counting the distance the ball traveled in both directions, up and down.
So, $400$ feet downward and $300$ feet upward adds up to $700$ feet.
answered Jul 23 at 13:01
David K
48.2k340107
Thank you so much! I feel kinda dumb now lol
– Suhani Shukla
Jul 23 at 13:06
add a comment |Â
up vote
2
down vote
If you ignore the first drop and the first rise, i.e. the first $175$ feet, what remains is three quarters of the total.
$$t-175=frac34timplies t=700.$$
answered Jul 23 at 13:07
Yves Daoust
111k665203
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
You counted only the distance the ball traveled downward on each bounce. The answer key is counting the distance the ball traveled in both directions, up and down.
So, $400$ feet downward and $300$ feet upward adds up to $700$ feet.
answered Jul 23 at 13:01
David K
48.2k340107
Thank you so much! I feel kinda dumb now lol
– Suhani Shukla
Jul 23 at 13:06
add a comment |Â
up vote
2
down vote
You counted only the distance the ball traveled downward on each bounce. The answer key is counting the distance the ball traveled in both directions, up and down.
So, $400$ feet downward and $300$ feet upward adds up to $700$ feet.
answered Jul 23 at 13:01
David K
48.2k340107
Thank you so much! I feel kinda dumb now lol
– Suhani Shukla
Jul 23 at 13:06
add a comment |Â
up vote
2
down vote
up vote
2
down vote
You counted only the distance the ball traveled downward on each bounce. The answer key is counting the distance the ball traveled in both directions, up and down.
So, $400$ feet downward and $300$ feet upward adds up to $700$ feet.
answered Jul 23 at 13:01
David K
48.2k340107
You counted only the distance the ball traveled downward on each bounce. The answer key is counting the distance the ball traveled in both directions, up and down.
So, $400$ feet downward and $300$ feet upward adds up to $700$ feet.
answered Jul 23 at 13:01
David K
48.2k340107
answered Jul 23 at 13:01
David K
48.2k340107
answered Jul 23 at 13:01
David K
48.2k340107
answered Jul 23 at 13:01
David K
48.2k340107
48.2k340107
Thank you so much! I feel kinda dumb now lol
– Suhani Shukla
Jul 23 at 13:06
add a comment |Â
Thank you so much! I feel kinda dumb now lol
– Suhani Shukla
Jul 23 at 13:06
Thank you so much! I feel kinda dumb now lol
– Suhani Shukla
Jul 23 at 13:06
Thank you so much! I feel kinda dumb now lol
– Suhani Shukla
Jul 23 at 13:06
add a comment |Â
up vote
2
down vote
If you ignore the first drop and the first rise, i.e. the first $175$ feet, what remains is three quarters of the total.
$$t-175=frac34timplies t=700.$$
answered Jul 23 at 13:07
Yves Daoust
111k665203
add a comment |Â
up vote
2
down vote
If you ignore the first drop and the first rise, i.e. the first $175$ feet, what remains is three quarters of the total.
$$t-175=frac34timplies t=700.$$
answered Jul 23 at 13:07
Yves Daoust
111k665203
add a comment |Â
up vote
2
down vote
up vote
2
down vote
If you ignore the first drop and the first rise, i.e. the first $175$ feet, what remains is three quarters of the total.
$$t-175=frac34timplies t=700.$$
answered Jul 23 at 13:07
Yves Daoust
111k665203
If you ignore the first drop and the first rise, i.e. the first $175$ feet, what remains is three quarters of the total.
$$t-175=frac34timplies t=700.$$
answered Jul 23 at 13:07
Yves Daoust
111k665203
answered Jul 23 at 13:07
Yves Daoust
111k665203
answered Jul 23 at 13:07
Yves Daoust
111k665203
answered Jul 23 at 13:07
Yves Daoust
111k665203
111k665203
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2860332%2fconvergence-of-geometric-series-factor-3-4-word-problem%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
It goes up and comes down on each bounce :)
– dbx
Jul 23 at 13:01
Just multiply $400$ by $2$ to count the up and down motion, and then subtract $100$ because the first bounce does not have a corresponding upwards motion (if that makes sense).
– RayDansh
Jul 23 at 13:05