Question : 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 of nw for any natural number n is modeled by the following function : A (n ) = 100 [1 - (1 - r/100) ^n ] , for 0 <,= r <= 100 a. Sketch the graph of A when r = 10. b. Evaluate lim n approaches to infinity A (n ) and interpret the results.
well, since (1-r/100) < 1, the limit is zero as n->∞.
So, as you'd expect, lim(A) = 100%
visit wolframalpha and enter
100 [1 - (1 - 10/100) ^n ]
for the plot.
Thanks Steve
To sketch the graph of A when r = 10, we will evaluate A(n) for different values of n.
a. Sketching the Graph of A when r = 10:
Step 1: Substitute r = 10 into the formula 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]
Step 2: Calculate A(n) for several values of n to create a table:
n | A(n)
---------
1 | 10
2 | 19
3 | 27.1
4 | 34.39
5 | 40.95
6 | 46.86
Step 3: Plot the points from the table on a coordinate plane.
The x-axis represents n, and the y-axis represents A(n).
Step 4: Connect the points with a smooth curve. Since the formula involves an exponential term, the curve will approach but never reach 100%. You should notice that the curve is decreasing, but the rate of decrease decreases as n increases.
b. Evaluating lim n approaches to infinity A(n) and interpreting the results:
To evaluate the limit as n approaches infinity, we need to find the value that A(n) approaches as n becomes larger and larger.
lim n approaches infinity A(n) = lim n approaches infinity 100 [1 - (0.9)^n]
As n approaches infinity, (0.9)^n approaches 0 since the exponential term becomes smaller and smaller.
Therefore, lim n approaches infinity A(n) = 100 [1 - 0] = 100
Interpretation: The limit of A(n) as n approaches infinity is 100. This means that as the thickness of the glass continues to increase (nw for larger values of n), the percentage of light absorbed by the glass approaches 100%. In other words, almost all of the light passing through the increasingly thicker glass is absorbed, leaving very little transmitted light.