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� oo A(n) and interpret the results.
a. To show that A(n) is an increasing function of n, we need to show that A(n+1) > A(n) for any natural number n.
Let's compare A(n+1) and A(n):
A(n+1) = 100 [1 - (1 - r/100)^(n+1)]
A(n) = 100 [1 - (1 - r/100)^n]
To simplify, let's expand the expressions:
A(n+1) = 100 - 100(1 - r/100)^(n+1)
A(n) = 100 - 100(1 - r/100)^n
Now, let's subtract A(n) from A(n+1):
A(n+1) - A(n) = 100 - 100(1 - r/100)^(n+1) - (100 - 100(1 - r/100)^n)
= 100 - 100(1 - r/100)^(n+1) - 100 + 100(1 - r/100)^n
= 100 - 100(1 - r/100)^(n+1) - 100 + 100(1 - r/100)^n
= -100(1 - r/100)^(n+1) + 100(1 - r/100)^n
To prove that A(n+1) > A(n), we need to show that A(n+1) - A(n) > 0:
-100(1 - r/100)^(n+1) + 100(1 - r/100)^n > 0
Since 0 < r < 100 and n is a natural number, (1 - r/100)^(n+1) < (1 - r/100)^n. Therefore, -100(1 - r/100)^(n+1) < -100(1 - r/100)^n.
Thus, we have:
-100(1 - r/100)^(n+1) + 100(1 - r/100)^n > 0
This shows that A(n+1) > A(n), which implies that A(n) is an increasing function of n if 0 < r < 100.
b. To sketch the graph of A when r = 10, we substitute r = 10 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]
Now we can plot the values of A(n) for different values of n on a graph.
c. To evaluate lim(n->∞) A(n), we substitute n = ∞ into the equation A(n) = 100 [1 - (1 - r/100)^n]:
lim(n->∞) A(n) = 100 [1 - (1 - r/100)^∞]
Since (1 - r/100) is between 0 and 1, as n approaches infinity, (1 - r/100)^n approaches 0.
Therefore, lim(n->∞) A(n) = 100 [1 - 0] = 100.
Interpretation: The limit of A(n) as n approaches infinity is 100, which means that as the thickness of the glass (nw) increases without bound, the percentage of light absorbed by the glass approaches 100%. This suggests that an infinite thickness of glass would absorb all the light passing through it.