Mrs. eaton's class is participating in the "box tops for education" campaign. on the first day, her class collected 2 tops. on the third day, her class collected 8 tops. let d represent each collection day and n represent the number of tops collected on that day. based on the situation, john claims the number of tops collected can be modeled by an exponential function. riley disagrees and claims the number of tops can be modeled with a linear function. what is the number of tops collected on the sixth day based on the exponential model? what is the number of tops collected on the sixth day based on the linear model?

Each colection day: D
Number of tops collected on that day: N

D1=1; N1=2
D2=3; N2=8

1) Linear model
N-N1=m(D-D1)

m=(N2-N1)/(D2-D1)
m=(8-2)/(3-1)
m=(6)/(2)
m=3

N-N1=m(D-D1)
N-2=3(D-1)
N-2=3D-3
N-2+2=3D-3+2
N=3D-1

when D=6:
N=3(6)-1
N=18-1
N=17

What is the number of tops collected on the sixth day based on the linear model?
The number of tops collected on the sixth day based on the linear model is 17.

2) Exponential model
N=a(b)^D
D=D1=1→N=N1=2→2=a(b)^1→2=ab→ab=2   (1)

D=D2=3→N=N2=8→8=a(b)^3→8=a(b)^(1+2)
8=a(b)^1(b)^2→8=ab(b)^2   (2)

Replacing (1) in (2)
(2) 8=2(b)^2
Solving for b:
8/2=2(b)^2/2
4=(b)^2
sqrt(4)=sqrt( b)^2 )
2=b
b=2

Replacing b=2 in (1)
(1) ab=2
a(2)=2
Solving for a:
a(2)/2=2/2
a=1

Then, the exponential model is N=1(2)^D
N=(2)^D

When D=6:
N=(2)^6
N=64

What is the number of tops collected on the sixth day based on the exponential model?
The number of tops collected on the sixth day based on the exponential model is 64