It is a beautiful sunny day at the fair. The U.V. index for this day is 8 or higher so sunscreen is a must. The effectiveness of sunscreen is indicated by the sunscreen protection factor (SPF). The higher the SPF number the less U.V. rays can penetrate to burn the skin. When the protection factor (SPF), s, is known you can determine the percent, p, of the sun’s ultraviolet rays that pass through it by using the following mathematical model:
P=100/S
a) What are the asymptotes for this function? Interpret the meaning of the asymptotes based on the scope of the problem.
b) Graph the function
c) The sunbathers on the beach were using a sunscreen with SPF of 35. What percent of the sun rays will pass through to skin?
a) Asymptotes are imaginary lines that a graph approaches but never touches. In this case, the asymptotes of the function P = 100/S are vertical lines at S = 0 and P = 0.
Interpreting the meaning of the asymptotes in the context of the problem, we can say that there is no effectiveness of sunscreen when the SPF (S) is 0, as the percentage of sun's ultraviolet rays (P) passing through would be infinite. On the other hand, when the SPF is very high, approaching infinity, the percentage of rays passing through (P) approaches 0, indicating maximum effectiveness in blocking U.V. rays.
b) To graph the function P = 100/S, we can plot some points and then draw a smooth curve. Here are a few points to get started:
S = 1, P = 100/1 = 100%
S = 5, P = 100/5 = 20%
S = 10, P = 100/10 = 10%
S = 20, P = 100/20 = 5%
Plotting these points and connecting them with a curve, the graph would look like a decreasing curve approaching the vertical asymptote.
c) If the sunbathers on the beach were using a sunscreen with SPF of 35, we can find the percent of sun rays that will pass through by substituting the value of SPF (S) into the function equation:
P = 100/S
P = 100/35
P ≈ 2.86%
So, approximately 2.86% of the sun rays will pass through to the skin when using a sunscreen with SPF of 35.
a) To find the asymptotes of the function, we need to consider the behavior of the function as S approaches positive infinity and as S approaches zero.
1. As S approaches positive infinity, P = 100/S approaches zero. Therefore, the horizontal asymptote is the line y = 0. This means that as the SPF value of the sunscreen approaches infinity, the percentage of the sun's ultraviolet rays that pass through the sunscreen approaches zero. In other words, a higher SPF provides greater protection against UV rays.
2. As S approaches zero, P = 100/S approaches infinity. However, in the context of this problem, SPF values less than or equal to zero do not make sense, so we can disregard this case.
b) To graph the function, we can plot a few points to get an idea of its shape. Choose some SPF values and calculate the corresponding percentages as follows:
- SPF = 10: P = 100/10 = 10%
- SPF = 20: P = 100/20 = 5%
- SPF = 30: P = 100/30 ≈ 3.33%
- SPF = 40: P = 100/40 ≈ 2.5%
- SPF = 50: P = 100/50 = 2%
Plotting these points on a graph, we can observe a general decreasing trend as SPF increases, approaching the horizontal asymptote at y = 0.
c) To determine the percent of the sun's rays that will pass through the skin using sunscreen with an SPF of 35, we can substitute S = 35 into the equation:
P = 100/S = 100/35 ≈ 2.86%
Therefore, approximately 2.86% of the sun's ultraviolet rays will pass through the skin when using sunscreen with an SPF of 35.