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.
Solution in details please
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 the comparison, let's divide both sides of the equation by 100:
A(n+1)/100 = [1 - (1 - r/100)^(n+1)]
A(n)/100 = [1 - (1 - r/100)^n]
Now, we can subtract the two equations:
A(n+1)/100 - A(n)/100 = [1 - (1 - r/100)^(n+1)] - [1 - (1 - r/100)^n]
(A(n+1) - A(n))/100 = [1 - (1 - r/100)^(n+1)] - [1 - (1 - r/100)^n]
Expanding the brackets:
(A(n+1) - A(n))/100 = 1 - (1 - r/100)^(n+1) - 1 + (1 - r/100)^n
Simplifying:
(A(n+1) - A(n))/100 = (1 - r/100)^(n+1) - (1 - r/100)^n
Now, let's focus on the right-hand side of the equation:
(1 - r/100)^(n+1) - (1 - r/100)^n
Using the property of exponents, we can rewrite this as:
[(1 - r/100)^n * (1 - r/100)] - (1 - r/100)^n
Factoring out the common factor, (1 - r/100)^n, we have:
(1 - r/100)^n * [(1 - r/100) - 1]
Simplifying further:
(1 - r/100)^n * (1 - r/100 - 1)
(1 - r/100)^n * (-r/100)
Since 0 < r < 100, -r/100 is negative. Therefore, (-r/100) is less than 0.
Now, let's consider the left-hand side of the equation, (A(n+1) - A(n))/100. Since we know that (-r/100) is less than 0, multiplying it by (1 - r/100)^n only makes it more negative.
Therefore, we can conclude that (A(n+1) - A(n))/100 is negative, which means that A(n+1) is less than A(n). Hence, 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 plug in different values of n into the equation A(n) = 100 [1 - (1 - r/100)^n], and plot the corresponding A(n) values on a graph.
Let's choose a few natural numbers for n and calculate the corresponding A(n) values:
n = 0: A(0) = 100 [1 - (1 - 10/100)^0] = 0
n = 1: A(1) = 100 [1 - (1 - 10/100)^1] = 9.09
n = 2: A(2) = 100 [1 - (1 - 10/100)^2] = 18.18
n = 3: A(3) = 100 [1 - (1 - 10/100)^3] = 27.27
n = 4: A(4) = 100 [1 - (1 - 10/100)^4] = 36.36
Plotting these points on a graph, we get a curve that starts from the point (0, 0) and steadily increases as n increases. The curve approaches but never reaches 100% as n approaches infinity.
c. To evaluate lim(n→∞) A(n), we need to find the limit of A(n) as n goes to infinity.
Let's consider the expression inside the square brackets in A(n):
1 - (1 - r/100)^n
As n approaches infinity, the term (1 - r/100)^n will tend to zero, assuming that 0 < r < 100. Therefore, the expression inside the square brackets will approach 1, resulting in:
lim(n→∞) A(n) = 100 [1 - 1] = 0
Interpretation: The limit represents the percent of light that is absorbed by a piece of glass with an infinitely increasing thickness. As n approaches infinity, the absorbed light approaches 0%, indicating that most of the light passes through the glass without significant absorption.