Respuesta :
Answer: 1) [tex]t(n)=0.6(2)^n[/tex]
2) [tex]f(x)=10(5)^x[/tex]
Step-by-step explanation:
1) Let the function that shows the thickness of the paper after n folds,
[tex]t(n) = ab^n[/tex] ---------(1)
Since, According to the question,
Initially the thickness of the paper = 0.6
That is, at n = 0, t(0) = 0.6
By equation (1),
[tex]0.6 = a(b)^0\implies 0.6 = a[/tex]
Hence the function that shows the given situation,
[tex]t(n) = 0.6 b^n[/tex] -----------(2)
Again when we fold the paper the thickness of the paper will be doubled.
Thus, at n = 1, t(1) = 1.2
By equation (2),
[tex]1.2 = 0.6 b^1\implies 2 = b[/tex]
Thus, the complete function is,
[tex]t(n) = 0.6 (2)^n[/tex]
2) Let the function that is passing through the points (-2, 2/5) and (-1,2),
[tex]f(x) = ab^x[/tex] ---------(1)
For f(x) = 2, x = -1
By equation (1),
[tex]2= ab^{-1}[/tex] ---------(2)
Also, For f(x) = 2/5, x = -2
Again, By equation (1),
[tex]\frac{2}{5}= a(b)^{-2}[/tex]
[tex]\implies \frac{2}{5}=ab^{-1}b^{-1}=2b^{-1}[/tex]
[tex]\implies \frac{2}{5}=\frac{2}{b}[/tex]
[tex]\implies 2b=10[/tex]
[tex]\implies b = 5[/tex]
By substituting this value in equation (2),
We get, a = 10
Hence, from equation (1), the function that is passing through the points (-2, 2/5) and (-1,2),
[tex]f(x) = 10(5)^x[/tex]
