1.5 Problem

Course subject(s) 1. Introducing Mathematical Modelling

In this practice problem you will derive a differential equation and complete a first modelling cycle. But first, the problem!


A daycare center for two- to four-year-olds has acquired a new outside playground with grass, plants and a sandbox. Summer is coming and everyone is looking forward to let the children play outside. However, then the staff starts to worry about sun protection. Can they let the children play outside in the middle of the day? Is the sun screen they use sufficient?

First, information is collected:

  • Sunburn is considered a long term health risk, especially in small children.
  • In this country, without protection, lighter skinned children can be in the sun on a summers day between noon and 3 PM for only 10 minutes before their skin reddens and starts to get sunburned. Darker skinned children can stay out somewhat longer (20 to 30 minutes), but also not indefinitely before they develop sunburn as well.
  • The daycare center uses sunscreen lotion with a Sun Protection Factor (SPF) of 30.  This means that if the cream is applied in a thickness of 2 mg per square cm skin, you can stay out in the sun 30 times as long as without sunscreen.
  • Most people apply a much thinner sunscreen layer than 2 mg per square cm skin. Most people apply the suncream unevenly, and with an averagethickness of only 0.5 mg per square cm.

The daycare center instructs the staff to apply the suncream carefully, generously and often, on all exposed skin of all the children when they play outside. But, to be on the safe side, they estimate that their layers are only 0.5 mg per square cm skin.

They first want to know how this thinner layer affects the Sun Protection Factor of their suncream. And then, they want estimates for how long the children can safely play outside midday on a summers day.

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Mathematical Modeling Basics by TU Delft OpenCourseWare is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Based on a work at https://online-learning.tudelft.nl/courses/mathematical-modeling-basics/.
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