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?
clearly s cannot be zero
and, as s gets huge, p -> 0
a. To find the asymptotes of the function, we need to determine the values of s that make the denominator equal to zero. In this case, the denominator is s.
Setting s equal to zero, we find that there are no real values for s that make the denominator zero. Therefore, there are no vertical asymptotes for this function.
However, there is a horizontal asymptote at y = 0. As s increases towards infinity, p will approach zero. This means that as the SPF value of sunscreen increases, the percentage of the sun's rays that pass through to the skin decreases, approaching zero. In other words, if the SPF is infinitely high, no UV rays will penetrate the skin.
b. To graph the function, we can use the given mathematical model p = 100/s. We can plot points by choosing specific SPF values and calculating the corresponding percentages of UV rays that pass through.
Let's choose a few SPF values:
- 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
Plotting these points on a graph, we can see that as SPF increases, the percentage of UV rays that pass through decreases.
c. If the sunbathers on the beach were using a sunscreen with SPF 35, we can use the mathematical model p = 100/s to calculate the percentage of UV rays that pass through.
Substituting s = 35 into the equation, we have p = 100/35 ≈ 2.857.
Therefore, approximately 2.857% of the sun's rays will pass through to the skin when using a sunscreen with SPF 35.