It is a beautiful sunny day at Canada’s Wonderland. The U.V. index for this day is 8, or high, so sunscreen is a must for all people. The effectiveness of sunscreen is indicated by the sunscreen protection factor (SPF). The higher the SPF number the fewer 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) The sunbathers on the beach were using a sunscreen with SPF of 35.
What percent of the sun’s rays will pass through to skin?
To answer these questions, we need to understand the mathematical model given: p = 100/s, where p represents the percent of the sun's ultraviolet rays that pass through the sunscreen, and s represents the SPF (sunscreen protection factor).
a) Asymptotes are vertical lines that a function approaches but never touches. In this case, we can say that the function p = 100/s has two asymptotes: one at s = 0 and the other at s = ±∞.
Interpretation:
- When s = 0, the SPF becomes zero, which means there is no sunscreen applied. In this case, the model p = 100/s results in division by zero, which is undefined. Therefore, the function approaches the y-axis but never touches it. This implies that without any sunscreen, 100% of the sun's ultraviolet rays will pass through.
- When s approaches ±∞, meaning the SPF becomes extremely large or small, the model p = 100/s approaches zero. Therefore, as the SPF increases or decreases significantly, the percentage of the sun's rays that pass through the sunscreen approaches zero. This signifies that with higher SPF values, a lower percentage of ultraviolet rays can penetrate the skin.
b) If the sunscreen used has an SPF of 35, we can substitute s = 35 into the formula p = 100/s to determine the percent of the sun's rays that will pass through the skin.
p = 100/35
p ≈ 2.86
Therefore, approximately 2.86% of the sun's ultraviolet rays will pass through the skin with the use of sunscreen with an SPF of 35.