Question

The table below shows the sale, in dollars, at Jacob's store over a period of five months:

Month: 1 2 3 4 5
Sale:1,000 1,050 1102.50 1157.63 1215.51

Did the number of people at Jacob's store increase linearly or exponentially? (4 points)

Linearly, because the table shows a constant percentage increase in sales per month

Exponentially, because the table shows a constant percentage increase in sales per month

Linearly, because the table shows that sales increase by an equal factor for an equal increase in months

Exponentially because the table shows and equal increase in sales for an equal increase in months

Answer 1

What is the percentage increase from Month1 to Month2?

Answer 2

I'm sorry but I'm terrible at math so I don't really know
How do you find that out?

Answer 3

I actually want to learn

Answer 4

is it 50?

Answer 5

Value \(x\) becomes \(y\).

Divide \(y\) by \(x\): \(\frac{1050}{1000} = 1.05\)
The increase if \(1.05 - 1\). (always subtract \(1\)).
For example, 1000->2000 means a \(\frac{2000}{1000} - 1 = 2-1=1 = \frac{100}{100} = 100\%\) increase.

Answer 6

*if = is

Answer 7

so would it be 5%?

Answer 8

Yes

Answer 9

Great!! So what do i do next?

Answer 10

You can compute all %increases, ......... it will always be 5%. (trust me).

So we are in front of a "constant % increase of sales" (not an increase in percentage!)

Answer 11

ok but one question that I have is what is the difference between a constant increase and an exponential increase?

Answer 12

(without searching…) from memory, LINEAR means: "two months pass, I win twice what I win in a month, whatever 2 months we are looking at."
Not the case here.

Answer 13

Exponential means: start anywhere with some value \(A\). After \(k\) months, the value is \(A(1+r)^k\) (where percentage is \(r\)).
That is the case here. You (or I) have checked that between any two value, the %increase is 5%. That's what had to be checked.

Answer 14

So linear is when the percentage doesn't change and exponential is when it does over a certain amount of time?

Answer 15

Not really.

Let's call the values \(s_1, s_2,\dots,s_5\).
We saw that the increase from \(s_1\) to \(s_2\) is 5%.
If it were linear, the increase from Month 1 to Month 3 would be \(+(2\times 5\%)\).

But, here, \(s_1\to s_2\) (+5%),
\(s_2\to s_3\) (+5%)
etc.
\(s_4\to s_5\) (+5%)

Since an increase of 5% from value X to Y can be written: Y = X (1+ 0.05), our sales verify this:
\[s_4 = s_3(1+0.05) = s_2 (1+0.05)(1+0.05) \\= s_1(1+0.05)(1+0.05)(1+0.05)=s_1(1+0.05)^3\].

It is exponential.

Answer 16

oh ok

Answer 17

So after finding this what do we do next?

Answer 18

An increase in \(r\%\) (written as a number between 0 and 1) means multiplying by a factor\((1+r\).

Answer 19

\()\).

Answer 20

There are two lines starting with "Exponentially".

One is wrong.

Answer 21

But one says constant and the other says equal

Answer 22

it's the following noun that counts: "constant percentage increase" or "constant increase in sales" ?

Answer 23

so it would be the second option

Answer 24

out of all 4

Answer 25

I think so too.

Answer 26

Awesome!! Thank You!!!

Answer 27

yw Fairy!