It is a beautiful sunny day at the fair. The U.V. index for this day is 8, or high, so sunscreen is a must for all the young folk. 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. Graph the function.
c. 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?
surely you can graph the function. You keep posting and waiting for someone elseto do your work.
since division by 0 is undefined, there is a vertical asymptote at s=0
as s gets huge, the fraction gets small, so there is a horizontal asymptote at p=0
for (c), just plug in 35 for s ...
a. To determine the asymptotes for the given function, we need to consider the range of possible SPF values. The SPF value represents the effectiveness of sunscreen, so it must be a positive number. However, in the context of this problem, SPF cannot be 0 or negative.
1. Horizontal asymptote: As SPF approaches infinity (i.e., SPF -> ∞), the percentage of sun's rays that pass through the sunscreen (p) approaches 0. This means that as the SPF value increases, the sunscreen becomes more effective in blocking the UV rays.
2. Vertical asymptote: As SPF approaches 0 (i.e., SPF -> 0), the percentage of sun's rays that pass through the sunscreen (p) approaches infinity (∞). This implies that if the SPF value is very low or zero, the sunscreen has no effect in blocking the UV rays, and the percentage of UV rays that pass through approaches infinity.
b. To graph the function p = 100/s, we can plot some points and draw a smooth curve passing through those points. We'll choose SPF values and calculate the corresponding percentage values (p). Let's choose SPF values of 1, 10, 20, 30, 40, and so on.
When SPF = 1, p = 100/1 = 100%
When SPF = 10, p = 100/10 = 10%
When SPF = 20, p = 100/20 = 5%
When SPF = 30, p = 100/30 = 3.33%
When SPF = 40, p = 100/40 = 2.5%
Plotting these points on a graph, we would observe that as SPF increases, p decreases, getting closer and closer to 0.
c. Based on the given SPF value of 35, we can calculate the percentage of the sun's rays that will pass through the skin using the formula p = 100/s.
When SPF = 35, p = 100/35 ≈ 2.86%
So, approximately 2.86% of the sun's ultraviolet rays will pass through the skin when using a sunscreen with SPF 35.