Light is absorbed when it passes through a glass window. If r% of light is absorbed by a glass with thickness w , then the percent of light that is absorbed by a piece of glass with a thickness nw of for any natural number n is modeled by the following function:
A(n) = 100 [1 - (1 - r/100)^n], f or 0 _< r _< 100
a. Show that is an increasing function of n if 0 < r < 100.
b. Sketch the graph of A when r = 10.
c. Evaluate limn�¨�‡ A(n) and interpret the results.
To prove that A(n) is an increasing function of n for 0 < r < 100, we need to show that the difference A(n+1) - A(n) is greater than or equal to zero.
a. Calculating A(n+1) - A(n):
A(n+1) - A(n) = 100[(1 - (1 - r/100)^(n+1)) - (1 - (r/100)^n)]
Expanding the brackets:
A(n+1) - A(n) = 100 [1 - (1 - r/100)^(n+1) - 1 + (r/100)^n]
Simplifying further:
A(n+1) - A(n) = 100 [-(1 - r/100)^(n+1) + (r/100)^n]
Now, we need to prove that this expression is greater than or equal to zero.
Considering that 0 < r < 100, 0 < r/100 < 1.
For any value of x between 0 and 1, (1 - x) is always positive.
Therefore, (1 - r/100) is positive, and for any natural number n, (1 - r/100)^(n+1) is also positive.
Additionally, (r/100)^n is positive since r/100 is between 0 and 1.
So, -(1 - r/100)^(n+1) + (r/100)^n is a difference of two positive values, which means it is greater than or equal to zero.
Hence, A(n+1) - A(n) is greater than or equal to zero for 0 < r < 100.
Therefore, A(n) is an increasing function of n.
b. To sketch the graph of A when r = 10, we need to substitute this value into the equation A(n) = 100 [1 - (1 - r/100)^n].
A(n) = 100 [1 - (1 - 10/100)^n]
= 100 [1 - (1 - 0.1)^n]
= 100 [1 - 0.9^n]
Plotting the graph on a coordinate plane with n on the x-axis and A(n) on the y-axis, we will see a continuous increasing curve starting from A(0) = 0 and approaching 100 as n increases.
c. Taking the limit as n approaches infinity (lim n�¨�‡ A(n)), we can interpret the results.
lim n�¨�‡ A(n) = lim n�¨�‡ 100 [1 - (1 - r/100)^n]
When n approaches infinity, (1 - r/100)^n approaches zero since the base is less than 1.
So, the expression becomes:
lim n�¨�‡ A(n) = 100 [1 - 0] = 100
The interpretation of this result is that as the thickness of the glass increases without bounds, the percentage of light absorbed by the glass also approaches 100%.