Question

Distance an object falls Assuming air resistance is negligible,
a small object that is dropped from a hot air balloon
falls 16 feet during the first second, 48 feet during the second
second, 80 feet during the third second, 112 feet during
the fourth second, and so on. Find an expression for the distance
the object falls in n seconds.

Answer 1

math series

Answer 2

Write out a table first

Answer 3

Perfect!

Let's graph them:
<1,16>, <2,48>,<3,80>,<4,112>

It's a straight line -

http://www.wolframalpha.com/input/?i= {{1%2C16}%2C+{2%2C48}%2C{3%2C80}%2C{4%2C112}}

So we can use y=mx + b.

Do you know how to find m and b?

Answer 4

Here's the chart -
http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427e58t8jopvv2

Answer 5

\[m =\frac{ \Delta Y}{ \Delta X }\]

Answer 6

SO R = the rate of the series
is the same as m ( the slope )

Answer 7

?

Answer 8

so you have the formula for m.

m=(112-16)/(4-1) = ?

Answer 9

yes . .

Answer 10

so once you have m, you have

y=mx+b

To find b you make x = 1 and y = 4:

4=m(1) + b
b = ?

That's it!

y = mx + b, where x is time and y is distance.

Answer 11

so you are saying that the expression of the series is the equation of a line ?

Answer 12

that's right!

Answer 13

i'd nver thought that would be the aproach

Answer 14

you don't need physics here

Answer 15

but what about the relationship is no linear ?

Answer 16

normally projectile motion is parabolic. But your numbers are describing a linear relationship if you see the graph

Answer 17

does this make sense?

Answer 18

well yeah i solved one of bacteria growth and n is a exponent

Answer 19

Chlorine levels Chlorine is often added to swimming pools
to control microorganisms. If the level of chlorine rises
above 3 ppm (parts per million), swimmers will experience
burning eyes and skin discomfort. If the level drops below
1 ppm, there is a possibility that the water will turn green
because of a large algae count. Chlorine must be added to
pool water at regular intervals. If no chlorine is added to a

Answer 20

during a 24-hour period, approximately 20% of the
chlorine will dissipate into the atmosphere and 80% will remain
in the water.
(a) Determine a recursive sequence (\[a _{n}\]) that expresses the
amount of chlorine present after n days if the pool has
ppm of chlorine initially and no chlorine is added.
(b) If a pool has 7 ppm of chlorine initially, construct a
table to determine the first day on which the chlorine
level will drop below 3 ppm.

Answer 21

this one is tough

Answer 22

has \[a _{0}\] ppm of chlorine*

Answer 23

You know, the 1st problem may actually be saying that the distance fell at each time period was IN ADDITION to that distance already traveled. And in that case, the relationship would be nonlinear.

I actually think that is how to interpret the problem.

So, we have

$$
d=at^2
$$

as our equation.

We find a by setting t=1 and d=16:

$$
a=16
$$

So our equation is

$$
d=16t^2
$$

So in 4 seconds,

$$
d=16\times4^2=256
$$

In 3 seconds,

$$
d=16\times 3^2=144
$$

The difference between 4 and 3 seconds is
256-144=112

Which is what was given.

Answer 24

well im going to work a little the problem

Answer 25

so in this case a is the rate ?

Answer 26

Are we still talking about the 1st problem?

Answer 27

yes

Answer 28

ok, yes, "a" is a parameter for a parabola. It's not necessarily a "rate". It's a number that says how "wide" the parabola is.

Answer 29

I'm starting to get confused you know why, i was expecting for the expression always behave as i want

Answer 30

i meant look at the d = 48

Answer 31

is an inequality

Answer 32

What do you mean it's an inequality? You mean it doesn't equal something?

Answer 33

48 = 16(2) ^2

Answer 34

ok.

so at t=2

d=16*2^2=16*4=64 feet

right?

Answer 35

d = t ^2 + 16 ?

Answer 36

$$
d=16\times t^2
$$

Answer 37

yeah but then i have to subtract 16 to give off 48, remember the list above

Answer 38

in second'th second the distance is 48

Answer 39

d represents the total distance the object falls, which was the question

To validate your table, you'll need to compute d for various t in your table and subtract them.

For example,

at t=2 seconds the total distance the object has fallen is

d=16*2^2=64

at t=3 seconds the total distance the object has fallen is

d=16*3^2=16*9=144

So to validate the information given and your table, you need to subtract these two numbers.

144-64=80.

Answer 40

this is getting hard, I'll take a look later of the whole process

Answer 41

so for the second problem i have to do the same process ?

Answer 42

The 1st problem is not hard but different in the sense that they give you differences and not totals. They tell you by how much something increases and so you need to infer that to get totals, you need to add to the previous results. That's the hardest part. Once you realize that this is a nonlinear function (i.e. you can't use y=mx+b) then the next option is quadratic (i.e. parabola). Then you need to ask, is the parabola wide? Does it shift upwards? does it shift left? and so on. If the curve had been more complicated, like multiple "bends", you would need to count the bends and add 1 to find the degree of the polynomial needed to represent the chart. All this comes with experience. Don't fret. Reread what we did here and get back with me on your questions.

Answer 43

well im familiar with functions and parabolas

Answer 44

but i thought it would be refer only to sequences and sum notation

Answer 45

cause that is what im studying right now

Answer 46

so i thought it would be a way or a formula to found the expression

Answer 47

to find*

Answer 48

The 2nd problem you need to write some raw numbers and find the pattern

Are you familiar with recursive sequences?

Answer 49

You could have approached the 1st problem recursively I think.

Answer 50

in each sequence we have \[(a,a_{n})\]

Answer 51

for each term

Answer 52

yeah recursively thta's what i need

Answer 53

well, I'm familiar but that not implies I'm good, at all

Answer 54

brb in about 15 mins

we'll solve the 1st one recursively

Answer 55

lol, man i start to realize, this is not meant for me

Answer 56

ok

Answer 57

did you get there ( I mean the solution)

Answer 58

Let's solve the 1st equation recursively:
$$
a_1=16\\
a_2=48\\
a_3=80\\
a_4=112
$$
Look for a pattern:
$$
a_1=16\times 1\\
a_2=16\times3\\
a_3=16\times 5\\
a_4=112\times 7
$$
Do you see the pattern? Look how the last term is related to the index of a:

$$
a_n=16\times(2n-1)
$$

Check: To find distance falls in 3 seconds after it has already fallen 2 seconds is
$$
a_3=16\times((2\times3)-1)=16\times(6-1)=16\times5=80
$$
Does this make sense?

Answer 59

*Correction on \(a_4\) above
$$
a_4=16\times 7
$$

Answer 60

\[a_{n+1} = a _{1} (2n+1) \]

Answer 61

that would be the expression fo the nth second

Answer 62

?

Answer 63

Let's check:
$$
a_1=16\\
a_2=a_1\times3\\
a_3=a_1\times 5\\
a_4=a_1\times7
$$
So,

$$
a_n=a_1\times(2n-1)
$$

Then
$$
a_{n+1}=a_1\times(2(n+1)-1)=a_1\times(2n+1)
$$
Which is what you got!

Answer 64

As you can see there are various ways to approach recursive problems and also different ways to solve this problem, generally.

Answer 65

what a relief ! what would i do if you weren't to help me out

Answer 66

Does that mean you understand?

Ready for the second problem?

Answer 67

oh are you serious ? well,
let's try to solve it

Answer 68

It's easy! Let's start.

Answer 69

Let X represent the amount of chlorine initially in the water in units of ppm.

After the 1st day (24 hours), there will only be

$$
a_1=0.8X
$$
ppm left.

Here we let \(a_n\) represent the amount of chlorine we have left after \(n \) days.

This is was the hardest part.

Make sense so far?

Answer 70

ok

Answer 71

first expression could be also \[a _{0}\] in ppm

Answer 72

yes, it could be. Multiple ways to start off.

Answer 73

$$
a_0=0.8X\\
a_1=0.8\times a_0=0.8^2\times X\\
a_2=0.8^3\times X\\
...\\
a_n=0.8^{n+1}X
$$
Or
$$
a_{n-1}=0.8^{n}X
$$

Is this ok?

If we had left \(a_1=0.8X\) then our answer would have been
$$
a_n=0.8^nX
$$
Which looks a little easier to ready.

But all these forms are equivalent.

Answer 74

I was thinking if there is no CL added in the nth days , at some point, we wouldn't find any Cl , at all

Answer 75

The amount of Cl will decrease steadly according to the equation above and will only approach zero ppm, but will never be zero but approaches it in the limit:

$$
\lim_{n\to\infty}0.8^nX=0
$$