Answer :
Sure! Let's tackle each part of the question step-by-step.
### Part (a): Write a function [tex]\( g(n) \)[/tex]
We want to determine the thickness of the paper after folding it [tex]\( n \)[/tex] times. Each fold doubles the thickness. If the initial thickness of the paper is 0.05 mm, after 1 fold the thickness is [tex]\( 2 \times 0.05 \)[/tex] mm. After 2 folds, it is [tex]\( 2 \times 2 \times 0.05 \)[/tex] mm, and so on.
To express this relationship as a function, we use:
[tex]\[ g(n) = 0.05 \times 2^n \][/tex]
where [tex]\( g(n) \)[/tex] is the thickness in millimeters after [tex]\( n \)[/tex] folds.
### Part (b): Write the inverse function [tex]\( g^{-1}(t) \)[/tex]
The inverse function [tex]\( g^{-1}(t) \)[/tex] helps us find the number of folds needed to reach a certain thickness [tex]\( t \)[/tex]. We start from the equation:
[tex]\[ t = 0.05 \times 2^n \][/tex]
To solve for [tex]\( n \)[/tex], we need to isolate it:
1. Divide both sides by 0.05:
[tex]\[ \frac{t}{0.05} = 2^n \][/tex]
2. Use the logarithm (base 2) to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \log_2\left(\frac{t}{0.05}\right) \][/tex]
So, the inverse function is:
[tex]\[ g^{-1}(t) = \log_2\left(\frac{t}{0.05}\right) \][/tex]
### Part (c): Determine the number of folds needed for a thickness equal to the distance from Earth to the Moon
The distance from Earth to the Moon is given as 384,472,300,000 mm. We apply the inverse function [tex]\( g^{-1}(t) \)[/tex] from Part (b) to find the number of folds needed:
[tex]\[ n = \log_2\left(\frac{384472300000}{0.05}\right) \][/tex]
By calculating the above expression, we find that:
[tex]\[ n \approx 42.81 \][/tex]
Since we cannot have a fraction of a fold, we'll need to fold the paper approximately 43 times for the thickness to reach or exceed the distance from the Earth to the Moon.
So, the number of folds needed is about 43 folds.
### Part (a): Write a function [tex]\( g(n) \)[/tex]
We want to determine the thickness of the paper after folding it [tex]\( n \)[/tex] times. Each fold doubles the thickness. If the initial thickness of the paper is 0.05 mm, after 1 fold the thickness is [tex]\( 2 \times 0.05 \)[/tex] mm. After 2 folds, it is [tex]\( 2 \times 2 \times 0.05 \)[/tex] mm, and so on.
To express this relationship as a function, we use:
[tex]\[ g(n) = 0.05 \times 2^n \][/tex]
where [tex]\( g(n) \)[/tex] is the thickness in millimeters after [tex]\( n \)[/tex] folds.
### Part (b): Write the inverse function [tex]\( g^{-1}(t) \)[/tex]
The inverse function [tex]\( g^{-1}(t) \)[/tex] helps us find the number of folds needed to reach a certain thickness [tex]\( t \)[/tex]. We start from the equation:
[tex]\[ t = 0.05 \times 2^n \][/tex]
To solve for [tex]\( n \)[/tex], we need to isolate it:
1. Divide both sides by 0.05:
[tex]\[ \frac{t}{0.05} = 2^n \][/tex]
2. Use the logarithm (base 2) to solve for [tex]\( n \)[/tex]:
[tex]\[ n = \log_2\left(\frac{t}{0.05}\right) \][/tex]
So, the inverse function is:
[tex]\[ g^{-1}(t) = \log_2\left(\frac{t}{0.05}\right) \][/tex]
### Part (c): Determine the number of folds needed for a thickness equal to the distance from Earth to the Moon
The distance from Earth to the Moon is given as 384,472,300,000 mm. We apply the inverse function [tex]\( g^{-1}(t) \)[/tex] from Part (b) to find the number of folds needed:
[tex]\[ n = \log_2\left(\frac{384472300000}{0.05}\right) \][/tex]
By calculating the above expression, we find that:
[tex]\[ n \approx 42.81 \][/tex]
Since we cannot have a fraction of a fold, we'll need to fold the paper approximately 43 times for the thickness to reach or exceed the distance from the Earth to the Moon.
So, the number of folds needed is about 43 folds.
