Question

The number N of aluminum cans used each year varies directly as the number of people P using the cans. If 55 people use 14,740 cans in one year, how many cans are used in a city which has a population of 1,383,000?

Answer 1

when things vary directly to each other; an increase in one increases the other; and a decrease in one decreases the other.

N = P in this case, but we need another term to justify that when N=14740 then P=55

N = kP

14740 = k(55)

14740/55 = k = 2948/11

Now we can determine any value of N or P based upon the constant of variation k

N = (2948/11)P

Answer 2

So when P = 1,383,000

N = (2948/11) (1383000) = 370,644,000 , if i hit the right keys lol

Answer 3

I am so lost right now. so N is the alluminum cans and p is the people where did the k come from to make n-KP?

Answer 4

"k" is the \(\color{red}{\text{"constant of variation"}}\).

We have two pieces of information;

1) we now that as N increases, P increases .... hence,
N varies DIRECTLY with P.

2) When P=55, we observe that N=14740

This gives us the relationship between N and P such that:

N = P
14740 = 55 ; but as is thats just a jibberish equality so we need
to calibrate it with some constant term to make
it sensible, and usable; well introduce the
"constant of variation" (k) into the mix to help
us out

14740 = k55 ; now what does k need to be for this to
make sense?

14740/55 = k = 2948/11 after reducing

What this means is the for ANY value of P we can determine the value for N such that

\[N=\frac{2949}{11}P\]

When the people(P) = 1,383,000 ; what will the number of cans(N) be?

\[N=\frac{2949}{11}(1,383,000)=370,644,000\]