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.
To show that A(n) is an increasing function of n, we need to show that the value of A(n+1) is greater than or equal to A(n) for any natural number n.
a. Let's consider two consecutive terms of A(n) and A(n+1):
A(n) = 100 [1 - (1 - r/100)^n]
A(n+1) = 100 [1 - (1 - r/100)^(n+1)]
We need to show that A(n+1) - A(n) ≥ 0.
A(n+1) - A(n) = 100 [1 - (1 - r/100)^(n+1)] - 100 [1 - (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) ≥ 0, we need to prove that -100(1 - r/100)^(n+1) + 100(1 - r/100)^n ≥ 0.
Dividing both sides by 100, we have:
-(1 - r/100)^(n+1) + (1 - r/100)^n ≥ 0
This can be rewritten as:
(1 - r/100)^n [(1 - r/100) - 1] ≥ 0
(1 - r/100)^n (1 - r/100 - 1) ≥ 0
(1 - r/100)^n (-r/100) ≥ 0
Since 0 < r < 100, -r/100 < 0. Also, (1 - r/100)^n ≥ 0 for any natural number n.
Therefore, the inequality holds, and A(n+1) - A(n) ≥ 0.
Since A(n+1) - A(n) ≥ 0, it means that A(n+1) is greater than or equal to A(n). This 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 need to evaluate A(n) for different values of n.
A(n) = 100 [1 - (1 - 10/100)^n]
= 100 [1 - (1 - 1/10)^n]
= 100 [1 - (9/10)^n]
We can calculate A(n) for various values of n to plot the graph. For example:
When n = 0, A(0) = 100 [1 - (9/10)^0] = 100 [1 - 1] = 0
When n = 1, A(1) = 100 [1 - (9/10)^1] = 100 [1 - 9/10] = 10
When n = 2, A(2) = 100 [1 - (9/10)^2] ≈ 19
When n = 3, A(3) = 100 [1 - (9/10)^3] ≈ 27
When n = 4, A(4) = 100 [1 - (9/10)^4] ≈ 34
Plotting these points on a graph will give us an increasing curve representing A(n) when r = 10.
c. To evaluate lim(n→∞) A(n) and interpret the results:
lim(n→∞) A(n) = lim(n→∞) [100 [1 - (1 - r/100)^n]]
Since the value of r/100 is between 0 and 1, as n approaches infinity, (1 - r/100)^n approaches 0. Therefore, the limit can be simplified to:
lim(n→∞) A(n) = 100 [1 - 0] = 100
The interpretation of this limit is that as the thickness of the glass approaches infinity, the percentage of light absorbed by the glass approaches 100%. This means that with an infinitely thick piece of glass, all of the light passing through it will be absorbed.
Note: The validity of this interpretation assumes the model given by A(n) is an accurate representation of light absorption in glass.