The exponential model can be written in the form y = ab^x, where a and b are constants.
To find the values of a and b, we can use the given points (2, 54.61) and (4, 403.48).
Let's start with point (2, 54.61). Substituting the x and y values into the exponential model equation, we get:
54.61 = ab^2
Next, let's use the second point (4, 403.48). Substituting the x and y values into the exponential model equation, we get:
403.48 = ab^4
We now have a system of equations:
54.61 = ab^2
403.48 = ab^4
To solve this system, we can divide the second equation by the first equation:
(403.48)/(54.61) = (ab^4)/(ab^2)
Simplifying, we get:
7.39 = b^2
Taking the square root of both sides, we find:
b ≈ √7.39
To find the value of a, we can substitute the value of b back into one of the original equations. Let's use the first equation:
54.61 = a(√7.39)^2
Simplifying, we get:
54.61 = a(7.39)
Dividing both sides by 7.39, we find:
a ≈ 54.61/7.39
Now we have the values of a and b:
a ≈ 7.39
b ≈ √7.39
So, the exponential model is approximately y = 7.39(√7.39)^x.
Now, let's move on to part b. To find the value of y when x = 8, we can substitute x = 8 into the exponential model equation:
y = 7.39(√7.39)^8
Calculating this expression will give us the value of y when x = 8.