I am looking for good motivational examples of how quadratic equations can naturally arise in real life for someone starting high school. The high school book my child is using just jumps into factoring and solving them without any clear purpose. The few motivations seem to be contrived, such as the flight of a ball. 

A related question I found is the following: [Everyday Example Problems for Solving Linear and Quadratic Equations](https://matheducators.stackexchange.com/questions/11310/everyday-example-problems-for-solving-linear-and-quadratic-equations). My question is different because I am looking for motivations.

A possible starter example: 

*A farmer has 2400 ft. of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? [1]*

The only concern I have here is that the quadratic function does come up but the solutions to the obvious quadratic equation are not immediately intuitive.

Ideally an example of how they arose in antiquity would be great.

1. <https://studentsuccess.asu.edu/sites/default/files/optimization.pdf>