Mrs. Eaton's class is in the "Box Tops for Education" campaign. On the first day, her class 2 tops. On the third day, her class collected 8 tops. Let D each collection day and N the of tops on that day.
Based on the situation, John claims the number of tops can be modeled by an exponential . Riley and claims the of tops can be modeled with a linear function. What is the of tops on the sixth day based on the model? What is the of tops on the sixth day based on the linear model?
Number of tops on 6th day based on exponential model:
Number of tops on the 6th day based on the linear model:
To determine the number of tops on the sixth day based on the exponential model, we can create an exponential function using the data points given:
Let D be the day and N be the number of tops.
From the given data:
D = 1, N = 2
D = 3, N = 8
We can write these as two points on an exponential function: (1, 2) and (3, 8).
Using these two points, we can find the exponential function:
N = a * b^D
2 = a * b^1
8 = a * b^3
Solving the first equation for a, we get a = 2 / b.
Substitute this into the second equation:
8 = (2 / b) * b^3
8 = 2 * b^2
b^2 = 4
b = 2
Now that we know b = 2, we can find a:
2 = a * 2
a = 1
Therefore, the exponential function is N = 2^D.
To find the number of tops on the sixth day based on the exponential model:
D = 6
N = 2^6 = 64
Therefore, based on the exponential model, the number of tops on the sixth day is 64.
Now, using the linear model:
To determine the number of tops on the sixth day based on the linear model, we can create a linear function using the data points given:
From the given data:
D = 1, N = 2
D = 3, N = 8
We can find the linear equation using these two points:
N = mx + b
Using (1, 2):
2 = m * 1 + b
m + b = 2
Using (3, 8):
8 = m * 3 + b
3m + b = 8
Subtracting the first equation from the second:
2m = 6
m = 3
Substitute m = 3 into the first equation:
3 + b = 2
b = -1
Therefore, the linear function is N = 3D - 1.
To find the number of tops on the sixth day based on the linear model:
D = 6
N = 3*6 - 1
N = 18 - 1
N = 17
Therefore, based on the linear model, the number of tops on the sixth day is 17.